INDUCTIVE  vs.  DEDUCTIVE  METHODS 
OF  TEACHING 


No.   11 

Inductive  versus  Deductive 

Methods  of  Teaching:  An 

Experimental  Research 

By 

W.  H.  WINCH 

External  Member  of  the  Psychological  Board  of  Studies  for  the  University  of  London 

Chairman  of  the  Committee  of  the  Teachers'  Guild  of  Great  Britain  and  Ireland 

on  Psychological  Research  in  Schools  ;  Lecturer  for  the  London  County 

Council  on  Pedagogical  Methods  in  Schools. 

Author  of '  'Problems  in  Education , "   ' ' German  Schools, "  "  When 
Should  a  Child  Begin  School,"  etc. 


. ».  A* 

WARWICK  &  YORK,  Inc. 
1913 


Copyright,  1913 
WARWICK    &   YORK.  Inc. 


EDITOR'S  PREFACE. 

It  affords  me  great  pleasure  to  call  editorial  atten- 
tion to  this  interesting  and  instructive  contribution 
to  experimental  pedagogy.  Mr.  Winch  writes  with 
the  authority  of  long  experience  born  of  his  profes- 
sional duties  as  one  of  the  official  inspectors  of  Eng- 
lish schools.  He  is,  indeed,  well  known  as  the  first 
Englishman  to  bring  the  technique  of  experimental 
and  statistical  methods  to  bear  upon  the  actual  prac- 
tical problems  of  the  school. 

Those  who  have  followed  with  any  care  the  modern 
developments  of  educational  theory  know  how  sig- 
nificant is  that  trend  of  investigation  which  seeks  to 
study  the  concrete  problems  of  education  at  first 
hand  in  the  classroom  and  with  all  the  exactness  of 
experimental  control.  The  movement  for  experi- 
mental pedagogy  is  yet  in  its  infancy,  but  it  has 
already  shown  the  possibilities  that  lie  before  it.  In 
the  Journal  of  Educational  Psychology,  with  which 
this  series  of  Educational  Psychology  Monographs 
is  affiliated,  there  has  appeared  of  late  an  important 
series  of  articles  which  show  for  various  school  sub- 
jects what  important  problems  offer  hope  of  solution 
by  experimental  investigation.  This  monograph 
presents  what  is  at  the  very  least  a  first  approxima- 


Z  '  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

tion  toward  the  solution  of  one  of  these  vexed  ques- 
tions of  educational  practice :  Is  it  better  to  follow 
deductive  or  inductive  methods  in  the  teaching  of 
various  types  of  subject-matter!  The  presentation 
has  the  special  merit  of  being  sufficiently  detailed 
that  any  teacher  who  desires  to  do  so  may  of  himself 
repeat  the  experiments  and  verify  the  conclusions. 

G.  M.  W. 


AUTHOR'S  PREFACE. 

This  is,  I  believe,  the  first  attempt  to  decide  be- 
tween the  conflicting  claims  of  ' inductive'  and  ' de- 
ductive '  methods  by  experimental  procedure.  In  the 
' world  of  science'  it  is  not  usual  for  results  to  be 
accepted  unless  the  methods  by  which  they  have  been 
obtained  are  described  in  such  detail  as  enables  other 
workers  to  repeat,  corroborate,  or  modify  them.  Nor 
are  they  regarded  as  valid  unless  they  are  obtainable 
under  widely  differing  external  circumstances.  To 
produce  similar  evidence  for  educational  science  will 
be  the  aim  of  all  serious  workers  in  education  during 
the  next  two  or  three  decades,  and  I  am  therefore 
offering  this  research  as  a  contribution  to  the  scien- 
tific knowledge  of  the  results  of  inductive  and  de- 
ductive methods  in  actual  application  under  school 
conditions. 

I  am  quite  well  aware  that  much  valuable  know- 
ledge is  collected  by  school  administrators  and  school 
inspectors  during  the  ordinary  course  of  their  work. 
They  know  much  about  the  results  of  the  application 
of  different  methods  in  different  schools.  But  to  dis- 
entangle all  the  contributory  factors — even  to  realize 
them — is  very  difficult,  and  inspectors  are  likely  to  be 
misled ;  for  the  teacher  is,  naturally,  mainly  desirous 
of  showing  that  his  school  is  a  good  one,  and  not  of 


4  INDUCTIVE   VS.    DEDUCTIVE    METHODS. 

settling,  by  experimental  tests,  the  value  of  a  par- 
ticular method.  The  work  reported  in  this  mono- 
graph is  not  subject  to  this  source  of  error,  since  the 
teachers,  in  this  case,  were  working  with  the  experi- 
menter, and  not  against  him.  It  is  my  firm  and  ever- 
growing conviction  that  without  that  kind  of  co-op- 
eration on  the  part  of  teachers  there  can  never  be,  in 
an  applicable  sense,  a  ' Science'  of  Education. 

W.  H.  W. 
London,  September,  1912. 


CONTENTS. 

Statistical  Note 7 

I.     Introduction 11 

II.     The  Problem  of  the  Experiments 17 

III.  The  General  Plan  of  the  Experiments 20 

IV.  First  Series  of  Experiments  : 

1.  General    Characteristics    of    the    Children    Who 

Worked  the  Exercises 23 

2.  The  Preliminary  Tests 23 

3.  The  Method  of  Marking  the  Preliminary  Tests      .  25 

4.  The  Teaching  of  the  Two  Groups 31 

5.  The  Immediate  Testing  of  the  Two  Groups    ...  35 

6.  The  Marking  of  the  Tests 36 

7.  The  Subsequent  Testing  of  the  Two  Groups  on  the 

Same  Subject-matter 37 

8.  The  Testing  of  the  Two  Groups  on  New  Material  .  38 

9.  The  Marking  of  the  New  Material 40 

10.  Chronology  of  the  Experiment 44 

11.  Results : 

(a)  The  Marks  for  the  Preliminary  Tests  ...  45 

(b)  The  Marks  for  the  Test  Immediately  After 

the   Definitions    Had    Been   Taught   and 

Learnt 46 

(c)  The  Marks  for  the  Tests  of  Deferred  Repro- 

duction       47 

(d)  Correlation    Between    Immediate    and    De- 

ferred Reproduction 49 

(e)  Results  When  the  Two  Groups  Are  Tested  on 

New   Material       50 

12.  Pedagogical  Conclusions 53 

V.     Second  Series  of  Experiments : 

1.  General  Plan 55 

2.  The  Preliminary  Tests  and  the  Method  of  Marking.  56 

3.  Chronology  of  the  Experiment 59 

4.  The  Final  Tests  and  the  Method  of  Marking      .     .  60 

5.  Results  of  the  Experiment : 

(a)  Results  of  the  Preliminary  Tests    ....  61 

(b)  Results  of  the  Tests  in  Immediate  and  De- 

ferred Reproduction 63 


(c)  Correlation    Between    Immediate    and    De- 

ferred Reproduction 65 

(d)  Results  of  the  Test  on  New  Material    ...    67 
VI.    Third  Series  of  Experiments : 

1.  General  Plan .69 

2.  The  Preliminary  Tests  and  the  Method  of  Marking.    70 

3.  Chronology  of  the  Experiment 75 

4.  The  Final  Tests  and  the  Method  of  Marking     .    .    76 

5.  Results  of  the  Experiments: 

(a)  Results  of  the  Preliminary  Tests      ....    90 

(b)  Results  of  the  Tests  in  Immediate  and  De- 

ferred Reproduction 92 

(c)  Correlation  Between  the  Results  of  Immedi- 

ate and  Deferred  Reproduction    ....    95 

(d)  Results  of  the  Test  on  New  Material    ...    96 
VII.    Fourth  Series  of  Experiments : 

1.  General   Plan 100 

2.  The  Preliminary  Tests  and  the  Method  of  Marking.  101 

3.  Chronology  of  the  Experiment      .......  104 

4.  The  Tests  of  Immediate  and  Deferred  Reproduc- 

tion       107 

5.  The  Test  of  Application  to  New  Material  ....  107 

6.  Results 114 

VIII.    Fifth  Series  of  Experiments: 

1.  General  Plan 119 

2.  The  Preliminary  Tests  and  the  Method  of  Marking.  120 

3.  Chronology  of  the  Experiment 122 

4.  The  Tests  of  Reproduction 124 

5.  The  Test  of  Application  to  New  Material  ....  129 

6.  Results : 

(a)  Of  the  Preliminary  Tests -   .  133 

(b)  Of  Immediate  Reproduction 134 

(c)  Correspondence  Between  Immediate  and  De- 

ferred Reproduction 135 

(d)  Results  of  the  Test  on  New  Material  ...  138 
IX.     General  Summary 140 


STATISTICAL  NOTE. 

Suppose  we  have  two  measurements  of  any  mental 
function  for  a  number  of  children,  that  the  second 
measurement  gives  higher  results  than  the  first  in 
most  cases,  and  that  the  average  mark  for  the  second 
measurement  is  a  little  higher  than  for  the  first. 
May  we,  therefore,  conclude  that  some  general  tend- 
ency is  at  work,  or  must  we  regard  the  higher  aver- 
age of  the  second  measurement  as  the  result  of 
4 chance'  or  mere  variability?  To  answer  this  ques- 
tion I  propose  to  illustrate  the  usual  statistical  check 
on  results  of  this  kind  by  means  of  one  or  two  exam- 
ples. Suppose  the  children  are  measured  for  their 
power  of  spontaneous  definition ;  that,  a  week  later, 
they  are  measured  again,  and  that  the  marks  are  as 
shown  in  the  following  table : 

r    First  Second 
Name.                                        measurement.          measurement 

A.  B 9  10 

C.  D 8  9 

E.  F 7  8 

G.   H 6  7 

I.  J 5  6 

K.  L 4  5 

M.  N 3  4 

O.   P 2  3 

R.   S 1  2 

Average,  5  Average,  6 

Common  sense  has  no  difficulty  in  deciding  that 
there  is  a  i general  tendency'  to  improvement  from 
one  exercise  to  another.  Let  us  now  calculate  the 


8  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

' probable  errors.'    The  ' probable  error'  of  an  aver- 

.    .67449(7      .  ,  .     .,  ,     ,   ,     .  ,. 

age  is ,  where    ^  is  the  standard  deviation, 

Vn 

and  'n'  is. the  number  of  cases  measured/  Worked 
out  on  this  formula,  the  '  probable  error'  of  the  aver- 
age 5  is  approximately  .6,  and  of  the  average  6  is  also 
approximately  .6.  The  ' probable  error'  of  the  dif- 

I       2 I          2 

ference  betiveen  two  averages  is  .67449  V-*7 


n 

where  'o^'  is  the  standard  deviation  of  the  first  aver- 
age, '<r29  is  the  standard  deviation  of  the  second  aver- 
age, and  'n'  is  the  number  of  cases  measured.  Ap- 
plying this  formula  to  the  present  example,  we  have 
the  *  probable  error'  of  the  difference  between  the 

two  averages  =  .67449  V  ,(2-6)2  +  (2-6)2?  which  is 

n 

approximately  .8. 

It  is  required  statistically  that  the  difference  be- 
tween two  means  shall  be  twice  (or  more)  the  i prob- 
able error'  of  that  difference  before  the  difference  is 
supposed  to  be  '  significant, '  that  is,  indicative  of  a 
general  tendency.  But  the  difference  between  the 
means  in  this  case  is  only  1  and  its  ' probable  error' 
is  .8,  so  that,  apparently,  we  have  no  ' significant' 
difference  at  all. 

But  let  us  consider  one  more  illustration  in  which 
the  averages  are  the  same,  but  in  which  common 
sense  would  not  find  a  general  tendency  to  improve- 
ment: 


*Simple  illustrations,  in  which  V  is  found  from  easy  examples, 
are  given  in  the  statistical  note  attached  to  my  monograph,  When 
Should  a  Child  Begin  School? 


STATISTICAL   NOTE.  9 

First  Second 
Name.                                        measurement.          measurement. 

A.  B 9  2 

C.  D 8  3 

E.  F 7  4 

G.   H 6  5 

I.  J 5  6 

K.  L 4  7 

M.  N 3  8 

O.   P 2  9 

R.   S 1  10 

Average,  5  Average,  6 

The  difference  between  the  means  is  again  equal 
to  1,  and  the  ' probable  error'  of  the  difference,  cal- 
culated just  as  before,  is  .8.  Statistically,  therefore, 
we  are  precisely  in  the  same  position  as  in  the  pre- 
vious example,  and  there  is  no  ' general  tendency7  to 
improvement.  But  quite  obviously  the  two  cases  are 
by  no  means  similar  and  their  i  probable  errors '  are 
not  the  same,  for  we  have  overlooked  the  positive 
correspondence  between  the  first  and  second  meas- 
urements of  A.  B.,  C.  D.,  and  the  rest  in  the  first 
illustrative  case  and  the  negative  correspondence  in 
the  second  illustrative  case.  The  theory  of  statistics 
takes  account  of  this  correspondence,  or  lack  of  cor- 
respondence, in  the  following  formula  for  the  '  prob- 
able error'  of  the  difference  between  two  averages: 


.67449 


n 


where  X'  is  the  standard  deviation  of  the  first  aver- 
age, *<r29  is  the  standard  deviation  of  the  second  aver- 
age, 'r'  is  the  coefficient  of  correlation  between  the 
two  series  of  measurements,  and  'n'  is  the  number  of 
cases  measured.  I  suggest  now  that  the  correlation 


10  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

coefficient  be  worked  out  for  both  cases.  In  the  first 
case  *r  '  will  be  found  to  be  +  1  and  in  the  second  case 
—  1.*  Let  us  then  apply  the  formula,  corrected  for 
correlation,  to  the  two  illustrative  cases.  The  '  prob- 
able error'  in  the  first  case  is 


(2.6)2  +  (2.6)2  -  2 


n 

It  will  be  seen  at  once  that  the  expression  disap- 
pears, for  2  (2.6)  (2.6)  =  (2.6)2  +  (2.6)2:  that  is  to 
say,  the  difference  between  the  averages  is  perfectly 
representative  of  the  two  series  of  measurements,  as 
common  sense  would  suppose.  In  the  second  case 
the  '  probable  error'  is 


.67449 


(2.6)2-2  (-1) 


n 


that  is,  double  what  it  was  when  the  negative  corre- 
lation was  neglected.  It  now  reaches  1.6,  and  is 
greater  than  the  difference  between  the  averages, 
which  is  only  1.  Hence  the  conclusion  is  against  any 
general  tendency,  again  in  accordance  with  common 
sense. 

These  illustrations  will  probably  be  sufficient  to 
show  that  the  use  of  probable  error  formulae  without 
regard  to  correlation  may  be  very  misleading,  and 
also  that  mere  averages,  without  some  indication  of 
the  nature  and  extent  of  the  variability  of  the  meas- 
urements, may  be  even  more  so. 


*Easy  illustrations  will  be  found  in  the  statistical  note  previously 
referred  to. 


INDUCTIVE  VERSUS  DEDUCTIVE  METHODS  OF 
TEACHING:  AN  EXPERIMENTAL  RESEARCH. 

I.    INTRODUCTION. 

In  England — it  is  for  Americans  to  speak  for  their 
own  country — there  is  a  widely-spread  opinion  that 
the  theory  and  the  practice  of  teaching  are  two  very 
different  things.  The  young  student  leaves  the  nor- 
mal school  or  training  college,  and,  doubtless  crudely 
enough,  begins  to  put,  or  to  try  to  put,  into  practice 
some  of  the  pedagogical  methods  which  he  has  been 
taught  as  theoretically  sound. 

Not  infrequently — I  had  almost  said  invariably — 
his  confidence  in  his  theoretical  instruction  receives 
a  violent  shock.  His  superiors  in  the  school  assure 
him  that  he  will  do  no  good  if  he  goes  on  like  that. 
What  is  worse  and  much  more  disconcerting  (for, 
after  all,  principals  and  head  masters  must  find 
fault;  it  is  their  metier),  his  confreres  look  on  with 
amused  tolerance  and  ' chaff'  him  about  his  'new- 
fangled' ways.  Then,  dropping  into  friendly  confi- 
dence, they  explain  to  him  that  it  was  all  very  well 
for  him  to  'get  up'  and  describe  methods  of  that  sort 
in  examination  papers ;  it  was  expected  of  him,  and, 
naturally  enough,  he  wished  to  get  his  certificate  of 

11 


12  INDUCTIVE   VS.    DEDUCTIVE    METHODS. 

proficiency  and  to  do  credit  to  his  college.  Exam- 
iners required  these  things;  they  were  unpractical 
persons  like  professors,  but,  of  course,  a  wise  student 
humored  them;  besides,  how  else  could  he  pass  his 
examinations  ?  Let  these  fellows  take  off  their  coats 
and  come  and  do  a  day's  teaching  in  the  schools,  and 
they  would  very  soon  change  their  opinions.  Their 
stuff  is  all  theory,  and  in  actual  school  life  is  simply 
no  good.  Now  you  have  become  a  man,  you  must  put 
away  from  you  childish  things ;  and  so  on.  Thus  the 
experienced  and  disillusioned  confreres  to  the  neo- 
phyte. 

It  is  not  clear  that  taking  off  their  coats  would 
assist  much  in  such  professional  conversions  as  are 
here  anticipated,  but  the  suggestion  is  a  protest 
against  what  the  teachers  regard  as  a  rather  vision- 
ary and  unpractical  existence. 

If  this  rude  shock  resulted  in  complete  divorce, 
there  would  be  some  hopes  of  other  and  happier  mar- 
riages between  theory  and  practice  later  on ;  but,  in 
England  at  least,  what  happens  is  rather  of  the  na- 
ture of  a  judicial  separation. 

The  theoretical  methods  are  not  absolutely  dis- 
carded; they  are  laid  by  and  put  in  evidence  only  on 
special  occasions ;  the  practical  methods  do  duty  day 
by  day.  For  it  is  dangerous,  from  the  standpoint  of 
professional  advancement,  for  the  teacher  boldly  to 
renounce  the  methods  he  has  been  taught ;  he  is  pur- 
sued all  his  life  by  unpractical,  theoretical  persons, 
to  wit,  inspectors,  and  he  will  often  deem  it  to  his 
advantage  to  profess  a  method  he  does  not  believe  in. 

Head  masters,  too,  mindful  of  the  repute  of  their 


INTRODUCTION.  13 

schools,  will  say,  "Yes,  that's  all  right,  but  don't  do 

that  when  Mr.  I r  comes ;  he  does  not  like  it ;  he 

thinks  you  ought  to  teach  that  this  way. ' ' 

Well,  yes,  no  doubt,  but  what  is  there  in  all  this 
but  the  usual  difficulty  which  besets  the  young  in- 
structed person  when  he  takes  an  actual  place  in  the 
working  world :  it  is  the  old  difficulty  of  science  ver- 
sus practice.  In  a  few  years  the  teacher,  like  other 
people,  will  have  allowed  his  theory  and  his  practice 
to  interpenetrate,  and  both  will  have  been  improved. 
In  such  wise  may  we  suppose  an  experienced  admin- 
istrator pooh-poohing  my  criticism. 

If  I  could  admit  this,  my  complaint  would  indeed 
lose  much  of  its  poigancy.  But  I  do  not  admit  it.  On 
the  contrary,  I  believe  that  with  the  great  majority 
of  teachers  there  remains  permanently  an  irrecon- 
cilable breach  between  dominant  theory  and  current 
practice.  It  is  true  that  experienced  teachers — 
some  of  them — will  attend  lectures  about  educational 
topics.  Two  men  speaking  somewhat  loudly  after 
leaving  a  lecture  hall — modesty  forbids  me  to  name 
the  lecturer — said  one  to  the  other:  "I  didn't  get 
much  from  him ;  he 's  like  all  the  rest  of  'em. "  ' '  Oh ! 
I  don't  know,"  said  the  other,  with  a  give-the-devil- 
his-due  air,  "one  gets  ideas."  "Yes,"  was  the 
prompt  reply,  "but  they  don't  work."  And  this,  let 
us  quite  clearly  understand  that,  is  not  merely  an 
expression  of  a  private  grumble;  it  is  a  strongly 
held  and  often  a  clearly  reasoned  view. 

There  are  always  "new  methods"  in  education,  of 
course,  and  I  hope  that  those  of  us  who  hold  the  very 
newest  of  them  are  more  or  less  prepared  to  see  our 
darlings  cold-shouldered  for  a  newer  birth.  Still, 
behind  all  temporary  fluctuations,  there  is  a  line  of 


14  ,       INDUCTIVE   VS.    DEDUCTIVE   METHODS. 

steady  meaning  in  such  phrases  as  'new  method/ 
' inductive  procedure,' ' developmental  method,' i psy- 
chological method,'  et  id  omne  genus.  And  behind 
all  temporary  oscillations  there  is  a  steady  trend  of 
opinion  amongst  experienced  teachers  that  these 
methods  have  certain  serious  disadvantages;  that 
though  they  may  be  valuable  for  show  purposes  in 
teaching,  they  are  too  slow,  and  the  information  thus 
acquired  is  not  really  available  when  it  is  wanted. 

An  experienced  head  master  in  London  wrote  to  a 
lecturer  on  pedagogy  in  the  following  terms : 

"I  (recently)  asked  a  question  on  the  difficulty  of 
covering  a  present  average  course  (by  using  the  new 
methods)  in  the  time  given  to  it  on  the  school  time- 
table, and  I  should  like  to  press  the  point  and  illus- 
trate its  importance  from  my  knowledge  of  the  views 
held  by  others,  and  especially  by  the  class  teachers  in 
my  own  school,  for  at  almost  every  conference  with 
my  staff  this  question  arises  strongly. 

"It  seems  impossible  to  train  children  by  much 
individual  work  in  class  by  inductive  methods,  much 
questioning  and  the  consequent  necessary  waiting 
for  the  child's  expression  to  be  formulated  in  a  suffi- 
ciently acceptable  form,  and  at  the  same  time  to  get 
through  the  course  set  in  a  given  time,  and  especially 
properly  to  prepare  the  children  also  for  examina- 
tion purposes. 

6 '  For  instance,  in  an  illustration  given  of  obtaining 
from  the  class  the  definition  of  the  diameter  of  a 
circle,  the  time  taken,  if  similarly  applied  to  other 
parts  of  the  course,  would  not  permit  of  a  present 
average  syllabus  being  more  than  about  half  com- 
pleted, nor  would  the  information  got  be  available 


INTRODUCTION.  15 

throughout  the  class  for  reproduction  at  a  more  or 
less  distant  examination. 

"Another  illustration :  Two  years  ago  an  inspector 
(fons  et  origo  malorum  W.  H.  W.),  examining 
Standard  V  (approximately  American  Grade  VI), 
asked  for  a  definition  of  a  proper  noun,  and,  not  get- 
ting a  satisfactory  answer,  tried  to  obtain  it  from  the 
boys  with  the  aid  of  many  questions  and  illustrations. 
He  took  up  twenty  minutes  of  the  lesson,  and  failed 
in  the  end  to  get  what  was  wanted. 

"Of  course,  all  the  time  the  children  were  being 
educated  on  the  best  lines;  they  showed  eagerness, 
interest  and  active  thought.  (This,  I  fear,  is  a  con- 
cession to  the  lecturer  as  a  theoretical  person.  W. 
H.  W.)  But,  the  time  taken,  in  view  of  the  rest  of 
the  syllabus,  was  excessive,  and  the  result  at  the  end 
was  not  satisfactory. " 

Then  follows  a  paragraph  in  which  the  writer  ex- 
presses his  willingness  to  carry  out  the  new  methods 
provided  the  educational  authority  will  dispense 
with  tangible  results. 

This  is  a  strong  letter.  It  expresses  views  which 
are  very  common,  and  which,  moreover,  are  held  by 
some  of  the  most  successful  schoolmasters  I  know. 
And  they  will  have  to  be  met  by  educational  science. 
I,  myself,  believe  that,  until  these  questions  are  dealt 
with  in  such  a  way  as  to  be  acceptable  to  teachers,  the 
breach  between  theory  and  practice  will  remain. 
Professors,  teachers  of  method  and  inspectors  may 
continue  as  now  to  receive  lip-homage  for  the  meth- 
ods they  advocate,  but  their  directions  will  be  hon- 
ored rather  in  the  breach  than  in  the  observance.  In 
actual  practice  there  will  be  little,  if  any,  change. 


16  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

What,  then,  do  I  suggest?  I  propose  the  experi- 
mental determination  of  these  disputed  questions  in 
the  schools  themselves.  There  is  an  increasing  num- 
ber of  teachers  willing — nay,  anxious — to  carry  out 
such  experiments  if  adequate  guidance  be  given  to 
them.  To  the  description  of  one  attempt  at  an  expe- 
rimental solution  of  some  of  these  disputed  points  I 
now  proceed. 


II.  THE  PEOBLEM  OF  THE  EXPERIMENT. 

No  one  can  hope  to  solve  all  the  questions  raised 
in  the  never-ending  controversy  about  inductive  and 
deductive  methods  by  means  of  a  single  experiment 
or  by  means  of  a  single  series  of  experiments.  Yet, 
if  one  attempts  to  deal  with  the  matter  experimen- 
tally, one  must  deal  with  some  definite  subject-mat- 
ter. There  is  danger  in  this,  since  we  may  find  out 
afterwards  that  our  conclusions  are  true  only  for 
subject-matter  of  that  particular  kind;  but  it  is  a 
danger  which  must  be  faced. 

A  good  plan,  if  one  is  conscious  of  bias,  is  to  select 
subject-matter  which  favors  the  chances  of  the 
method  in  which  personally  one  does  not  believe.  So 
I,  believing  in  inductive  rather  than  in  deductive 
methods,  chose  geometrical  definition  as  the  subject- 
matter  for  my  experiment. 

There  is  much  good  opinion  in  favor  of  deductive 
treatment  of  definitions,  especially  when,  as  in 
demonstrative  geometry,  a  sort  of  system  of 
reasoned  conclusions  is  to  be  built  up  upon  those 
definitions.  It  is  argued  that  a  pupil  ought  to  know 
exactly  what  the  definition  means,  that  the  exact 
wording  of  it  is  very  important  for  that  purpose, 
and  that  at  some  stage  in  the  procedure  the  defin- 
itions should  be  memorized.  There  is  no  question 
here  about  the  introduction  to  demonstrative  geom- 
etry. It  is  supposed  by  both  parties  to  the  dispute 

17 


18  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

that  manual  work  and  geometrical  construction  are 
necessary  propaedeutics  to  any  rational  system  of 
geometry.  But  if  we  are  ever  to  have  a  system  of 
demonstrative  geometry  we  must,  it  is  said,  have 
exact  meanings  for  our  terms  or  we  shall  never  be 
able  to  reason  in  words  at  all. 

This  is  by  no  means  a  weak  case,  and  to  it  addi- 
tional importance  is  given  at  the  present  juncture, 
when  so  much  dissatisfaction  is  being  expressed  as 
to  the  *  chaos'  into  which  geometrical  teaching  is 
falling  through  what  is  alleged  to  be  an  excess  of 
inductive  method. 

Those  who  advocate  purely  inductive  methods 
urge  that  the  memorizing  of  the  definition  and  the 
study  of  its  application  to  examples  is  not,  in  the 
truest  sense,  knowing  the  definition ;  it  is  urged  that 
it  can  be  known  better  if  it  is  built  up  from  the  ex- 
amples. It  is  asserted  that  the  memorizing  of  defini- 
tions leads  to  bad  errors,  '  howlers,'  of  which  the 
following,  once  given  to  me,  is  a  choice  example :  "  A 
circle  is  a  figure  bounded  by  a  straight  line,  which  is 
such  that  every  point  within  it  is  at  equal  distance 
from  every  other  point,"  There  is  a  tendency  to 
concede  that  inductive  methods  are  slow;  there  is 
some  tendency  also  to  concede  the  point  that  induc- 
tive methods  will  not  prepare  a  pupil  so  well  for 
examination  purposes.  But  it  is  argued  that  what 
he  does  learn  he  will  remember  longer,  and  that  he 
will  be  made  more  intelligent. 

The  use  of  the  word  *  intelligent'  in  educational 
disputes  amounts  almost  to  a  public  scandal,  and  I 
do  not  propose  to  use  it  without  giving  an  opponent 
some  way  of  checking  the  assertion. 

By  intelligence,  in  this  case,  I  am  going  to  mean 


THE   PEOBLEM   OF   THE   EXPEBIMESTT.  19 

the  power  gained  to  deal  with  new  material  in  con- 
sequence of  the  mental  processes  which  the  pupil  has 
passed  through  in  acquiring  the  old.  I  have  found 
this  interpretation  of  the  word  acceptable  to  both 
parties  in  the  dispute. 

Let  me  now  endeavor  to  disentangle  the  threads 
and  see  how  far  the  assertions  made,  first  on  one  side 
and  then  on  the  other,  are  susceptible  of  experimen- 
tal determination. 

First  of  all,  we  can  quite  easily  find  out  whether 
pupils  taught  inductively  or  deductively  know  the 
required  definitions  better  immediately  after  they 
have  acquired  them.  We  shall  demand  exact  mean- 
ings, but  not  a  stereotyped  form  of  words.  A  child 
taught  inductively  would  not,  of  course,  know  a  par- 
ticular form  of  words  for  a  definition,  but  no  devia- 
tion from  accuracy  of  meaning  must  be  allowed. 

Secondly,  we  can  find  out,  by  repeating  the  exer- 
cises later  on,  whether  the  pupil  forgets  more  or  less 
after  one  method  than  after  the  other.  We  are  thus 
testing  the  durability  of  his  knowledge. 

Thirdly,  we  can  find  out  how  many  'bad  errors' 
are  made  by  pupils  who  are  taught  by  inductive  and 
deductive  methods,  respectively. 

Fourthly,  we  can  find  out  which  of  the  two  methods 
gives  the  pupil  the  greater  power  of  attacking  new 
material  successfully. 

In  so  far  as  we  can  determine  these  issues,  we  are 
in  possession  of  the  clues  which  will  lead  us  to 
reasonable  conclusions  on  most,  if  not  all,  of  the 
questions  raised  in  this  section  and  in  the  section 
entitled  Introduction. 


III.    THE  GENERAL  PLAN  OF  THE  EXPERI- 
MENT. 

One  difficulty  in  all  work  of  this  kind  is  to  find 
some  unsophisticated  material  with  which  to  experi- 
ment. I  wanted  to  work,  if  I  could,  with  some  school 
children  who  had  never  learnt  or  even  heard  of  a 
geometrical  definition  throughout  the  whole  of  their 
school  lives.  I  think  I  succeeded  in  getting  this  con- 
dition fulfilled  with  some  Standard  V  girls  in  the 
southwest  of  London  and  some  Standard  III  boys  in 
the  southeast. 

In  the  course  of  the  experiment  one  of  the  boys' 
fathers  told  him  it  was  Euclid,  which  it  wasn't,  and 
gave  him  one  or  two  ' tips'  which  spoiled  some  of  his 
papers ;  but,  with  that  exception,  nothing  transpired 
to  indicate  that  we  were  not  working  on  virgin  soil. 
I  worked  also  with  a  Standard  VI  and  VII  girls' 
class  in  a  poor  school.'  These  girls,  though  not  au 
courant  with  geometrical  definition,  had  nevertheless 
done  much  constructive  work  and  were  accustomed  to 
express  themselves  freely  and  exactly.  I  worked  also 
with  a  Standard  VII  boys'  class  in  a  poor  school. 
There  was  a  little  difficulty  here  with  one  or  two  of 
the  definitions,  owing  to  the  boys'  attendance  at  the 
Manual  Training  Center,  where  they  had  learnt 
something  about  them.  And,  finally,  the  work  was 
done  with  an  ex- VII  boys'  class,  the  highest  class  of 
a  higher  grade  school.  The  teacher  of  this  class  was 

20 


GENERAL   PLAN    OF    THE   EXPERIMENT.  21 

a  man  who  had  for  years  attended  lectures  on  psy- 
chology, and  was  accustomed  to  teach  very  largely 
by  inductive  methods. 

I  think  it  will  be  conceded  that  we  have  here  a  good 
variety  of  material;  and,  since  in  every  case  we 
worked  with  all  the  pupils  of  the  class,  and  not 
merely  with  selected  pupils,  it  will  also  be  conceded 
that  if  any  tendencies  show  themselves  throughout 
the  whole  range  of  our  material,  the  probability  that 
they  are  chance  results  is  very  small  indeed.  So 
much  for  a  general  survey  of  the  material;  it  re- 
mains to  be  seen  how  it  was  utilized.  Briefly,  though 
there  were  local  variations  in  procedure,  the  same 
general  plan  was  followed  throughout. 

A  whole  class,  under  one  teacher  and  working  un- 
der the  same  syllabus  of  instruction,  was  divided  into 
two  equal  groups.  The  children  were  required  to 
attempt  the  definition  of  some  geometrical  shapes 
which  were  placed  before  them  in  the  form  of  large 
drawings,  and  on  the  results  of  these  attempts  the 
class  was  divided.  One  of  the  two  groups  subse- 
quently acquired  the  definitions  inductively.  The 
other  group  learnt  them,  but  the  children  were  in- 
structed that  the  exact  words  given  them  in  the  defi- 
nition were  not  required  as  long  as  they  gave  all  the 
meaning.  The  two  groups  were  tested  immediately, 
and  after  shorter  and  longer  intervals.  And,  after 
some  interval  of  time,  new  material  of  a  somewhat 
analogous  kind  was  given  to  the  children  to  define 
with  a  view  to  discovering  which  of  the  two  groups 
could  better  apply  their  old  knowledge,  as  we  say, 
though  it  is,  in  some  aspects,  rather  an  application  of 
a  method  than  an  application  of  knowledge. 

Two  of  the  classes  were  taken  by  me  in  the  indue- 


22  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

tive  acquisition  of  the  definitions ;  in  the  three  other 
cases  they  were  taken  by  their  own  teachers. 

In  one  of  the  schools  in  which  I  took  the  exercises 
myself  one  of  the  other  teachers  inquired  of  the 
teacher  of  the  class  whether  I  wished  the  inductive 
method  to  succeed.  ' '  I  think  he  does, ' '  was  the  reply. 
"Then,"  came  the  prompt  rejoinder,  "he  ought  to 
have  let  you  take  it ;  you  would  have  got  it  into  them 
much  better  than  he  would."  With  this  unsolicited 
testimonial  to  my  handling  of  the  method,  I  proceed 
to  a  detailed  description  of  the  five  series  of  experi- 
ments actually  carried  out  in  the  schools  above  cited. 


IV.    FIEST  SEEIES  OF  EXPERIMENTS. 

1.    General    characteristics    of    the    children    ivho 
worked  the  exercises. 

This  experiment  was  carried  out  during  the  months 
of  September,  October  and  November  of  the  year 
1911.  The  work  was  done  with  the  whole  of  a  Stand- 
ard V  class  of  girls  of  an  average  age  of  11  years  8 
months.  The  children  knew  nothing  about  geomet- 
rical definitions  and  were  not  biased  by  practice  or 
novelty  towards  either  of  the  methods  employed.  It 
is,  of  course,  necessary  to  know  the  customary  lines 
of  teaching  in  a  school  in  order  to  prevent  one  from 
drawing  misleading  conclusions  from  the  results  of 
experiments  of  this  kind.  The  infant  school  work 
which  these  girls  had  done  some  four,  five  or  six 
years  previously  was  very  little  affected  by  tend- 
encies to  the  kindergarten  or  sensory  type  of  instruc- 
tion. The  school  was  situated  in  a  neighborhood 
slightly  above  the  average  for  municipal  elementary 
schools,  and  the  girls  of  the  class  were  accustomed 
to  give  full  attention  to  their  school  work. 

{' 
2.    The  Preliminary  Tests. 

Drawings  of  squares,  triangles,  oblongs  and  diam- 
eters of  circles  were  placed  upon  large  blackboards, 
with  their  names  written  above  them,  thus: 

23 


24 


INDUCTIVE   VS.    DEDUCTIVE    METHODS. 


of 


FIRST    SERIES   OF   EXPERIMENTS.  25 

The  children  had  the  drawings  pointed  out  to 
them,  with  the  accompanying  words,  "  These  are 
squares, ?  9  etc.  In  the  case  of  the  last  set  of  figures 
the  straight  line  was  pointed  to  with  the  words, 
'  '  This  is  a  diameter  of  a  circle,  and  this  is  a  diameter 
of  a  circle,"  etc.  Then  upon  a  blackboard  the  fol- 
lowing questions  were  written : 

1.  What  is  a  square  ? 

2.  What  is  a  triangle? 

3.  What  is  an  oblong? 

4.  What  is  a  diameter  of  a  circle  ? 

All  the  children  in  the  class  were  required  to  an- 
swer the  questions.  They  were  told  to  do  so  thought- 
fully and  without  hurry.  No  time  limit  was  imposed 
in  this  or  in  any  other  of  the  subsequent  exercises  of 
this  experiment. 

3.     The  Method  of  Marking  the  Preliminary  Tests. 

The  importance  of  this  question  merits  a  short  dis- 
cussion of  the  principles  involved.  I  suppose  one's 
first  notion  is  something  like  this:  Let  us  just  take 
any  currently  accepted  definition  of  a  square,  etc., 
Euclidean  or  other,  and  mark  the  children's  exer- 
cises in  accordance  with  their  correspondence  or  non- 
correspondence  with  that.  It  will  not  matter  very 
much  which  definition  we  take,  provided  that  we  keep 
to  the  same  one  all  through  the  marking. 

A  perusal  of  the  answers  shows  us  immediately,  if 
we  had  not  known  it  before,  that  this  method  will 
not  do.  The  children  will  hardly  be  likely,  for  ex- 
ample, to  write  down  that  a  square  is  a  rectilinear 
quadrilateral  with  a  right  angle,  as  one  good  defini- 
tion gives  it.  They  know  the  meaning  of  a  straight 


26  INDUCTIVE   VS.    DEDUCTIVE   METHODS. 

line ;  they  know  the  meaning  of  four  sides ;  they  do 
not  know  the  meaning  of  right  angle,  and  if  they  did 
they  would  tell  you  quite  candidly,  if  they  had  been 
properly  taught,  that  the  definition  just  given  was 
wrong.  A  square  has  four  right  angles,  not  one,  they 
would  urge.  The  teacher  might  quite  authoritatively 
inform  them  that  every  four-equal-sided  straight- 
lined  figure,  if  it  have  one  right  angle,  must  have 
four;  therefore,  why  say  four  in  the  definition?  and 
that  a  definition  should  not  say  more  than  it  need. 
I  hope  the  teacher  won't,  because  the  redundancy 
is  no  redundancy  to  the  child  at  this  stage ;  indeed,  is 
no  redundancy  at  all  until  the  child  is  in  a  mental 
condition  to  deduce  some  of  the  properties  from  the 
others.  Then,  and  only  then,  can  one  strike  out  the 
derivable  properties  and  be  satisfied  with  the  others 
as  sufficiently  defining  the  term.  This  is  an  exceed- 
ingly interesting  exercise,  but  its  time  is  not  yet. 
4  *  Well,"  I  can  hear  an  impatient  mathematician  say, 
"this  isn't  mathematics;  this  is  psychology;  the  chil- 
dren are  not  going  to  be  marked  on  real  definitions 
at  all. ' '  On  the  contrary,  I  assert  that  they  are  going 
to  be  marked  on  the  very  realest  of  definitions ;  they 
will  be  marked  according  to  the  number  of  qualities 
and  properties  which  they  can  themselves  see  to  be 
common  to  all  the  specimens  called  by  the  same  name. 
And  I  go  further,  and  assert  that  the  mathema- 
tician's definition,  suitable  and  right  for  those  who 
know  the  system  of  knowledge  within  which  the  defi- 
nition finds  a  place,  is  just  mere  arbitrariness  to 
those  who  do  not.  And,  moreover,  to  tell  the  child 
that  he  may  mention  some  of  the  common  properties 
that  he  finds,  and  that  he  may  not  mention  others,  un- 
less he  is  in  a  position  to  see  for  himself  that  some 


FIRST   SERIES   OF   EXPERIMENTS.  27 

are  derivatives  or  derivable,  is  to  shut  him  up  sharp 
before  a  mystery.  He  won't  do  much  spontaneous 
defining  after  that.  Perhaps,  some  day,  we  may  have 
a  system  of  demonstrative  geometry  built  up  by  psy- 
chological research.  If  so,  the  definitions  will  change 
as  knowledge  accumulates  and  reasoning  becomes 
more  penetrating,  just  as  our  definitions  do  of  the 
common  objects  of  daily  life.  Imagine  a  system  of 
geometry  for  school  children  with  evolving  defin- 
itions. It  is  doubtless  too  horrible,  and  I  do  not,  at 
present,  ask  my  reader  to  accept  such  a  thing,  but 
only  to  grant  that  if  I  want  to  mark  fairly  the  efforts 
of  untaught  children  in  spontaneous  definition  I  must 
be  guided  by  what  they  do,  and  not  mark  on  an  a 
priori  scheme,  settled  beforehand  by  Euclid  or  by 
another  geometer,  or  by  myself,  who  am  no  geometer. 

What,  then,  do  the  children  say  in  answer  to  these 
questions,  "What  is  a  square?'7  etc.  There  are  be- 
fore us  between  forty  and  fifty  sets  of  answers,  and, 
though  it  would  be  illuminating  for  anyone  to  read 
the  whole  of  them,  they  cannot  be  reproduced  here. 
The  interest  in  them  lies  in  the  fact  that  they  repre- 
sent untaught,  spontaneous  attempts ;  it  is  an  inter- 
est which  is  at  first  psychological;  the  pedagogical 
interest  comes  later.  Perhaps  one  of  the  best  papers 
and  one  of  the  poorer  ones  may  be  found  worthy  of 
attention.  I  request  the  reader  to  look  at  the  draw- 
ings whilst  considering  the  children's  definitions. 
The  specimen  papers  follow  exactly  as  they  were 
written : 

Nellie  W.,  aged  12  years  3  months,  wrote : 

1.  A  square  is  an  object  with  four  corners  and  four  lines,  two 
for  the  sides  and  one  for  the  top  and  bottom. 

2.  A  triangle  is  an  object  with  three  corners,  and  three  lines, 
two  slanted  ones  and  one  straight  one  at  the  bottom. 


28  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

3.  An  oblong  is  an  object  with  four  corners  and  four  lines,  the 
lines  at  the  top  and  bottom  being  longer  than  those  of  the  sides. 

4.  A  diameter  of  a  circle  is  a  line  drawn  right  across  the  circle 
from  one  side  to  the  other  each  line  is  called  a  diameter. 

Winnie  T.,  aged  12  years  8  months,  wrote : 

1.  A  square  is  a  kind  of  box  with  four  lines  all  the  same  length. 

2.  A  triangle  is  a  thing  with  three  sides  not  all  the  same  length. 

3.  An  oblong  is  a  thing  with  two  short  sides  and  two  long  sides. 

4.  A  diameter  is  a  circle  with  a  number  of  lines  going  from  one 
side  to  the  other. 

Even  from  these  two  papers  one  may  get  a  hint  as 
to  the  way  the  children  are  going  to  sum  the  figures 
up,  and  a  careful  search  through  all  the  papers  re- 
veals that  by  one  or  another  the  following  points  of 
accurate  definition  are  mentioned : 

Common  qualities   mentioned  in  children's   defini- 
tions. 

1.  A  square  is  a  shape,  figure,  drawing. 
It  has  lines  or  sides. 

It  has  four  lines  or  sides. 

It  has  equal  lines  or  sides. 

It  has  straight  lines  or  sides. 

It  has  corners. 

It  has  four  corners. 

It  has  equal  corners. 

(A  total  of  eight  points.) 

2.  A  triangle  is  a  shape,  figure,  drawing. 
It  has  lines  or  sides. 

It  has  three  lines  or  sides. 

It  has  corners. 

It  has  three  corners. 

(A  total  of  five  points.) 


FIRST   SERIES   OF   EXPERIMENTS.  29 

3.  An  oblong  is  a  shape,  figure,  drawing. 
It  has  lines  or  sides. 

It  has  four  lines  or  sides. 
It  has  straight  lines  or  sides. 
It  has  two  long  sides. 
It  has  two  short  sides. 
The  two  long  sides  are  the  same  length. 
The  two  short  sides  are  the  same  length. 
The  two  long  sides  are  opposite  each  other. 
The  two  short  sides  are  opposite  each  other. 
It  has  corners. 
It  has  four  corners. 
The  corners  are  all  the  same  size.) 
(A  total  of  thirteen  points.) 

4.  A  diameter  of  a  circle  is  a  line. 
It  is  a  straight  line. 

It  goes  through  the  middle  or  center  of  a  circle. 
It  goes  from  one  side  or  edge  of  the  circle  to 
the  other. 

(A  total  of  four  points.) 

The  papers  were  marked  thus :  One  mark  was  al- 
lowed for  each  common  attribute  correctly  given.  It 
was  decided  not  to  allow  thing  or  object  as  equivalent 
to  shape,  diagram,  etc.,  for  it  was  thought  that ' thing' 
was  so  wide  a  term  as  to  be  hardly  available  for  the 
purpose  of  these  definitions,  and  that  by  the  word 
' object'  children  usually  meant  a  material  thing  in 
three  dimensions  of  space. 

It  is  not  contended  that  all  the  above  units  of 
marking  are  exactly  equal  in  value ;  it  is  contended 
only  that  marking  on  these  units  is  easy,  practically 
serviceable,  and  yields  results  from  which  one  can 
draw  reliable  conclusions  for  practical  purposes. 


30  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

On  the  results  of  this  marking  all  the  children  of 
the  class  were  divided  into  two  equal  groups,  one  of 
the  best  children  being  placed  at  the  head  of  the  first 
group — Group  A — an  equivalent  child  being  placed 
at  the  head  of  the  second  group — Group  B — then  the 
children  next  in  order  were  placed,  one  in  each  group, 
and  so  on  down  the  list,  carefully  preserving  the  bal- 
ance all  the  way  down,  till  all  the  children  were  di- 
vided between  the  two  groups. 

There  is  a  weakness  here  which  needs  attention. 
It  is  not  usually  satisfactory  to  divide  a  class  on  the 
basis  of  one  test  only.  It  is  probably  much  more  sat- 
isfactory where  the  higher  mental  functions  are  con- 
cerned than  it  is  where  simple  sensory  functions  are 
measured,  but  it  is  risky  even  in  the  former  case.  It 
adds  enormously  to  the  probability  of  a  valid  result 
if  several  tests  of  the  same  kind  are  taken  and  the 
results  of  these  correlated.  One  then  feels  confi- 
dence, if  the  results  of  the  tests  correlate  highly  with 
one  another,  that  one  is  testing  some  function  or 
group  of  functions  which  is  operating  steadily,  and 
that  each  child  is  working  at  about  its  'true  form'  as 
compared  with  the  others.  In  some  of  the  subse- 
quent experiments  of  this  kind  I  adopted  that  plan, 
but  as  I  wished  to  be  present  during  the  whole  time, 
and  on  each  occasion  when  exercises  were  done  in 
this  case,  I  reduced  the  number  of  Preliminary  Tests 
to  one,  but  I  did  so  with  a  full  consciousness  that  I 
should  feel  less  reliability  in  the  equality  of  my  two 
groups. 

The  marks  obtained  in  the  Preliminary  Tests  by 
the  two  groups,  respectively,  will  be  shown  in  the 
section  headed  'Besults.' 


FIRST   SERIES   OF   EXPERIMENTS.  31 

4.    The  Teaching  of  the  Two  Groups. 

About  a  week  later  the  two  groups  were  separately 
instructed  in  the  subject-matter  of  the  definitions. 
As  each  child  had  already  made  some  attempts  for 
herself  under  test  conditions  to1  define  the  terms,  she 
was  in  a  favorable  condition  for  the  reception  of 
knowledge  by  any  method. 

I  had  decided  that  one  of  the  two  groups  should 
have  the  definitions  written  out  for  them,  with  illus- 
trative drawings  underneath,  and  that  the  children 
of  this  group  should  be  instructed  to  study  and  learn 
the  definitions.  They  were  told  that  they  were  going 
to  be  examined  afterwards,  that  they  might  then 
write  down  the  exact  words  if  they  liked,  but  that  as 
long  as  they  got  down  all  the  meaning  they  would 
lose  no  marks  because  they  had  failed  to  remember 
the  exact  words.  This  resembled  the  way  in  which  I 
remember  myself  to  have  learnt  the  Euclidean  defi- 
nitions. I  studied  the  words  and  let  my  attention 
pass  from  words  to  figures  to  see  the  illustrations  of 
the  general  statements  in  the  definitions — it  may 
fairly  be  called  a  deductive  method.  There  is,  how- 
ever, one  important  difference.  The  definitions  given 
to  the  children  are  such  as  they  themselves  are  capa- 
ble of  making  up.  That  does  not  mean  that  every 
child,  nor  even  any  child,  in  the  class  could  say  pre- 
cisely all  these  things  by  itself;  it  does  mean  that 
they  are  on  the  lines  of  the  child's  own  evolutionary 
track. 

The  Definitions  as  Learnt  Deductively. 

They  were  written  down  with  illustrative  examples 
drawn  underneath  the  definitions.  The  drawings 
were  the  same  as  those  given  before  in  the  Prelimi- 


32  INDUCTIVE   VS.    DEDUCTIVE   METHODS. 

nary  Tests ;  they  will  not  be  reproduced  here.    The 
definitions  were  worded  thus : 

1.  A  square  is  a  shape  with  four  sides  and  four 
corners.    The  sides  are  straight  and  they  are  all  of 
the  same  length;  the  corners  are  all  of  the  same  size. 

2.  A  triangle  is  a  shape  with  three  sides  and  three 
corners.* 

3.  An  oblong  is  a  shape  with  four  straight  sides 
and  four  corners.    Two  sides  are  long  and  two  are 
short.    The  two  long  ones  are  opposite  to  each  other 
and  are  of  the  same  length,  and  the  two  short  ones 
are  opposite  to  each  other  and  are  of  the  same  length. 
All  the  corners  are  of  the  same  size. 

4.  A  diameter  of  a  circle  is  a  straight  line  going 
through  the  middle  of  a  circle  from  one  side  of  the 
circle  to  the  other. 

The  Definitions  as  Taught  Inductively. 

The  teacher,  myself  in  this  case,  having  the  points 
of  each  definition  in  mind,  taught  up  to  them,  but  no 
instruction  was  given  by  the  teacher  otherwise  than 
by  questioning. 

In  another  school,  in  which  also  I  did  the  necessary 
teaching  myself,  a  discussion  arose  afterwards  be- 
tween the  two  groups  of  girls  because  one  group  had 
done  rather  better  than  the  other  in  the  subsequent 
testing.  "You  ought  to  do  best,"  said  a  girl  of  the 
deductive  group  to  the  inductive  group;  "the  gentle- 
man taught  you  the  definitions ;  we  had  to  learn  'em 
for  ourselves." 

"That's  just  where  you're  wrong,"  she  was  an- 


*It  will  be  remembered  that  one  of  the  triangles  drawn  had 
curvilinear  sides. 


FIRST    SERIES   OF   EXPERIMENTS.  33 

swered;  "the  gentleman  never  told  us  a  thing;  we 
told  him  all  about  it. ' ' 

But  perhaps  a  little  more  explicitness  may  be  use- 
ful to  those  who  may  wish  to  repeat  the  experiment. 
Let  me  illustrate  by  means  of  the  last  example : 

Pointing  to  all  the  diameters  drawn,  the  teacher 
says:  "What  can  we  say  about  all  these?"  The  an- 
swer ' lines'  will  be  received.  He  can  then  ask  the 
question:  "What  is  a  diameter  of  a  circle? "  He 
will  be  answered,  if  he  chooses  his  questionee  well: 
"A  diameter  of  a  circle  is  a  line."  One  feature  of 
the  method  is  that  the  teacher  accepts  all  the  answers 
given  to  him  and  translates  them  into  concrete  form. 
He  draws  a  curved  line  on  the  blackboard,  but  not 
within  one  of  the  circles,  and  asks :  "Is  that  a  diam- 
eter of  a  circle?"  He  is  answered:  "No,  because  it 
is  not  a  straight  line. ' '  He  draws  a  straight  line,  still 
outside  the  circle,  and  asks:  "Is  that  right?"  The 
answer  comes:  "No,  because  it's  not  in  a  circle." 
"Very  well,"  the  teacher  says,  "let  us  try  again. 
What  is  a  diameter  of  a  circle?"  If  he  chooses  a 
child's  answer,  as  he  should,  from  among  the  least 
proficient  of  the  class,  he  will  be  answered:  "A  di- 
ameter of  a  circle  is  a  straight  line  inside  a  circle." 
He  accepts  the  answer  and  draws  a  straight  line  in 
a  circle  which  neither  passes  through  the  center  nor 
touches  the  circumference  on  either  extremity.  He 
again  asks:  " Is  that  right ?"  He  will  be  told :  "No, 
because  the  line  does  not  go  to  the  edge  on  both 
sides."  He  corrects  his  drawing,  producing  the  line 
each  way  to  the  circumference.  He  will  now  be  told 
his  line  is  wrong,  because  it  must  pass  through  the 
middle  or  center  of  the  circle.  He  then  draws  a  fresh 
line  passing  through  the  center,  but  cutting  the  cir- 


34  INDUCTIVE   VS.   DEDUCTIVE    METHODS. 

cumference,  and  he  is  now  told  the  line  must  reach 
the  edges,  but  not  pass  over  them.  At  this  stage  he 
can  rely  upon  receiving  a  correct  definition  from  the 
great  majority  of  his  pupils ;  but  it  is  essential,  if  we 
are  to  keep  this  method  distinct  from  the  other  one, 
that  he  does  not  ask  a  number  of  children  to  give  the 
correct  definition.  One  or  two  may  do  so,  and  the 
teacher  then  passes  on;  otherwise  the  mnemonic 
repetitive  factor  comes  in  here  as  in  the  other 
method,  and  for  the  purposes  of  this  experiment  it  is 
usually  desirable  to  avoid  this. 

I  do  not  propose  to  go  in  detail  over  the  method 
employed  for  teaching  the  remaining  definitions. 
Any  experienced  teacher — and  this  paper  is  not  writ- 
ten for  other  than  experienced  teachers — will  be  able 
to  ask  analogous  questions  and  get  the  answers  cor- 
rected in  analogous  ways.  With  children  taught  fre- 
quently on  this  method  it  is  quite  possible  to  get  the 
necessary  drawings  and  corrections,  or  most  of  them, 
done  by  members  of  the  class,  so  that  the  machinery 
of  correction  and  amplification  is  mainly  in  the  hands 
of  the  class,  with  the  teacher  there  to  see  fair  play 
and  direct  the  discussion  to  profitable  issues.  But  I 
do  not  press  this  latter  point ;  the  work  of  concrete 
exemplification  of  error  was,  in  the  case  of  all  the 
experiments  about  to  be  described,  solely  in  the 
hands  of  the  teacher.  Teachers  who  would  not  agree 
with  the  method  of  mutual  correction  may  quite  well 
agree  with  this. 

There  are  one  or  two  points  of  detail,  however, 
which  may  cause  difficulty.  It  is  of  little  use  for  the 
teacher  to  point  to  the  squares  drawn  and  ask, "  What 
are  all  these?"  for  he  will  naturally  be  answered, 
" Squares."  Indeed,  the  word  squares  is  written 


FIRST   SEKIES   OF   EXPERIMENTS.  35 

above  the  drawings.  But  if  he  points  to  the  squares, 
and  the  triangles,  and  oblongs  as  well,  and  asks  the 
same  question,  he  will  get  answers  like  " drawings," 
"shapes,"  "figures,"  "diagrams."  He  can  then 
start  his  detailed  questioning  to  bring  out  the  defini- 
tion of  a  square.  Among  his  answers  will  very  likely 
be,  "A  square  is  a  shape  with  four  lines  and  four  cor- 
ners. "  It  is  obvious  that  many  figures  which  are  not 
squares  can  be  drawn  to  comply  with  this  definition, 
and  the  correction  will  proceed  as  before.  The  size 
of  a  corner  is  a  difficulty  for  young  children;  they 
confuse  corner  with  edge.  It  helps  to  ask,  (pointing 
to  angles  of  different  sizes)  "If  these  were  the  cor- 
ners of  a  room,  how  much  sand  or  how  many  blocks 
could  I  put  in  that  corner,  and  in  that  one  ? "  In  some 
such  way  the  size  of  the  corners  becomes  thinkable  to 
the  young  child.  The  question  has  been  raised 
whether  items  of  considerable  difficulty,  like  this  one, 
should  not  carry  more  than  one  mark.  The  theoret- 
ical justification  is  conceded,  but  it  is  argued  that  in 
practice  a  mark  for  each  unit  gives  sufficiently  steady 
and  reliable  results. 

Children  will  often  give  a  quality  which  is  true  of 
only  one  or  two  of  the  squares  or  triangles.  It  is  only 
necessary  to  point  to  the  other  ones  in  these  cases. 

I  need,  perhaps,  hardly  say  that  these  children,  like 
the  others,  knew  they  were  going  to  be  examined  im- 
mediately afterwards  on  the  work  they  were  then 
doing. 

5.    The  Immediate  Testing  of  the  Two  Groups. 

As  soon  as  the  teaching  of  the  Inductive  Group  was 
completed,  the  group  which  had  been  learning  the 
definitions  in  another  room  also  stopped  their  work ; 


36  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

and  in  a  third  room,  so  that  the  environment  of  both 
the  groups  should  be  changed,  both  sets  of  pupils 
answered  the  following  questions : 

1.  "What  is  a  square? " 

2.  "What  is  a  triangle f" 

3.  ' '  What  is  an  oblong  I ' ' 

4.  "What  is  a  diameter  of  a  circle?" 


6.     The  Marking  of  the  Tests. 

The  papers  were  marked  exactly  as  in  the  case  of 
the  Preliminary  Tests,  so  far  as  the  positive  units 
were  concerned,  but  a  new  feature  was  added  to  the 
marking.  It  is  well  known  that  bad  teaching  and  bad 
learning  produce  errors,  and  errors  of  a  noxious  kind. 
But  some  statements  that  we  sometimes  call  errors 
in  the  work  of  school  children  are  rather  irrelevances 
and  redundancies  than  errors.  For  instance,  when  a 
child,  in  defining  a  square,  after  mentioning  the  prop- 
erties of  a  square  quite  correctly,  says:  "and  the 
corners  are  opposite  each  other,"  the  statement  is 
worth  no  positive  mark,  but  neither  is  it  worth  any 
negative  mark.  Or  when  a  child,  in  defining  a  tri- 
angle, says :  ' '  Some  of  the  lines  are  curved  and  some 
are  straight,"  though  this,  strictly  speaking,  is  no 
part  of  the  definition  (which  includes  only  the  quali- 
ties common  to  all  the  triangles  given),  yet  it  can 
hardly  be  called  a  bad  error.  But  it  is  a  bad  error 
for  a  child  to  say:  "A  triangle  is  a  shape  with  three 
equal  lines  and  three  corners."  Such  an  answer  re- 
ceives five  positive  marks — one  each  for  shape,  lines, 
three  (lines),  corners,  three  (corners).  But  'equal' 
receives  a  negative  mark  as  a  'bad  error.7  Again, 
when  a  child,  in  defining  a  square,  says,  amidst  much 


FIRST   SERIES   OF   EXPERIMENTS.  37 

which  is  correct:  "The  corners  all  come  under  each 
other,"  the  statement  is  marked  as  a  bad  error.  Or 
when  a  child,  speaking  of  a  diameter,  says :  "It  must 
stand  upright,"  the  statement  is  regarded  as  a  bad 
error.  The  first  diameter  in  the  drawings  was  up- 
right, hence  the  error.  The  definition  was  being 
elaborated  from  a  memory  of  the  one  drawing  with- 
out comparison  with  the  memories  of  the  others. 

In  the  case  of  the  experiment  in  this  school,  there- 
fore, besides  giving  the  positive  marks  obtained  by 
each  group,  I  shall  also  give  the  negative  marks,  and, 
in  addition,  the  positive  marks  with  the  negative 
marks  subtracted  from  them. 

It  is  interesting,  before  turning  to  the  section  show- 
ing the  results,  to  try  to  guess  from  our  general 
knowledge  of  children's  minds  of  the  given  ages 
(roughly  from  eleven  to  twelve),  and  our  opinions 
as  to  the  methods  of  teaching  and  learning,  which  of 
the  two  groups  gained  the  more  positive  marks  and 
which  group  made  the  more  bad  errors. 

7.    Subsequent  Testing  of  the  Two  Groups  on  the 
Same  Subject-Matter. 

In  discussions  among  teachers  the  question  is  fre- 
quently raised  as  to  the  relation  between  the  quick- 
ness and  the  permanence  of  knowledge.  Teachers 
are  prone  in  theory  to  back  the  tortoise  rather  than 
the  hare,  though  in  practice  they  repeatedly  prod 
the  tortoise  up.  How  far  does  the  present  experi- 
ment throw  light  on  the  matter?  Are  we  justified  in 
supposing  that  a  test  given  to  two  groups  of  children, 
immediately  after  certain  knowledge  has  been  ac- 
quired, supplies  us  with  comparative  results  which 


38  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

will  be  true,  say,  a  week  later,  a  month  later,  and 
so  on? 

To  test  this  point  the  exercise  above  described  was 
repeated  a  Week  later.  The  children  were  not  aware 
that  they  would  ever  have  to  do  this  work  again. 

Then,  once  again,  more  than  a  month  after  the  first 
test  (the  exact  chronology  of  the  experiment  will  be 
given  later  on),  the  test  was  repeated  a  second  time. 
The  papers  were  marked  in  both  these  cases  exactly 
as  in  the  first  test,  positive  marks  being  given  for  the 
points  remembered  and  negative  marks  for  the  bad 
errors.  The  results  will  enable  us  to  see  how  far  the 
comparisons  between  the  groups  based  upon  the  im- 
mediate results  are  corroborated  when  the  results 
for  deferred  reproduction  are  taken  into  considera- 
tion. Again,  it  is  worth  while  to  try  to  think  the 
answer  out  on  general  principles  before  turning  to 
the  actual  results. 

8.    The  Testing  of  the  Two  Groups  on  New  Material. 

It  will  be  remembered  that  the  children  of  the  class 
were  divided  on  the  results  of  a  test  in  which  they 
were  required  to  find  definitions  for  themselves  of  a 
square,  etc.  How  far  has  the  teaching  or  learning 
by  different  methods  affected  their  power  to  attack 
new  material  of  an  analogous  kind?  This  is  one  of 
the  most  important  questions  that  can  be  asked  of 
any  method  of  teaching  or  learning. 

In  ordinary  pedagogical  discussions  it  would  be 
implied  by  assertions  that  children  would  be  made 
more  intelligent  by  one  method  than  by  another.  To 
investigate  this  point  experimentally  the  following 
tests  were  made : 

Drawings  were  shown  of  rhombuses,  etc.,  and  their 
names  written  above  them,  thus : 


FIRST   SEBIES   OP   EXPERIMENTS. 


39 


In  the  drawings  actually  used  the  diagonals  were  continuous 
lines  drawn  in  red. 


40  INDUCTIVE   VS.    DEDUCTIVE    METHODS. 

Then  the  following  questions  were  written  on  the 
blackboard : 

1.  "What  is  a  rhombus!" 

2.  ' '  What  is  a  trapezium  ? ' ' 

3.  <  *  What  is  a  rhomboid ! ' ' 

4.  "What  is  a  diagonal  of  a  square?" 

and  the  children  were  required  to  answer  them  in 
writing. 

9.     The  Marking  of  the  New  Material. 

As  in  the  case  of  the  Preliminary  Tests,  we  must 
look  for  the  basis  of  our  marking  in  the  papers  them- 
selves, and  not  in  any  a  priori  scheme  of  values.  It 
will,  I  think,  be  profitable  if  one  or  two  samples  of 
the  children's  actual  answers  be  given  before  con- 
sidering the  units  of  marking  which  were  adopted. 

Laura  B ,  aged  13  years  11  months,  who 

worked  in  the  deductive  group,  wrote : 

1.  A  rhombus  is  a  shape  with  four  sides  and  four  corners,  the 
four  sides  slant  the  same  way,  and  the  corners  are  the  same  size. 

2.  A  trapezium  is  a  shape  with  four  sides  and  four  corners. 
The  four  sides  are  not  the  same  length  and  do  not  slope  the  same 
way.    The  corners  are  not  the  same  size. 

3.  A  rhomboid  is  a  shape  with  four  sides  and  four  corners.    It 
has  two  long  sides  and  two  short  sides,  the  long  ones  are  opposite 
each  other,  and  the  short  ones  are  opposite  each  other,  the  corners 
are  opposite  each  other  and  slant  the  same  way,  the  corners  are 
all  the  same  size. 

4.  A  diagonal  of  a  square  is  a  shape  with  four  straight  lines 
and  four  corners,  the  lines  are  all  the  same  length  and  the  corners 
are  all  the  same  size  with  a  line  going  through  the  square  from  one 
corner  to  the  opposite  corner. 

Ehoda   T ,    aged   12   years    11   months,   who 

worked  with  the  inductive  group,  wrote : 

1.  A  rhombus  is  a  shape.     It  is  made  up  of  four  sides.    The 
lines  are  straight,  but  are  drawn  slantling,  and  there  are  four  cor- 
ners joined  exactly  to  one  another. 

2.  A  trapezium  is  a  shape.    It  is  made  up  of  four  sides  and  are 


FIRST    SERIES    OF    EXPERIMENTS.  41 

not  the  same  length.    The  lines  are  straight  but  are  drawn  slant- 
ling.    There  are  four  corners,  they  join  together  exactly. 

3.  A  rhomboid  is  a  shape.    It  is  made  up  of  four  sides  two  short 
sides  being  opposite,  and  the  t\vo  long  sides  the  same.    The  lines 
are  straight  but  are  drawn  slantling.    There  are  four  corners  they 
join  exactly  to  one  another. 

4.  A  diagonal  of  a  square  is  not  a  shape.    It  is  a  straight  line, 
but  drawn  slantling.     It  joins  two  corners  exactly  opposite  one 
another,  and  the  line  must  not  reach  over  the  two  corners. 

Even  from  these  two  papers  alone  it  would  not  be 
very  difficult  to  make  out  a  scheme  of  marking,  and 
when  taken  in  conjunction  with  the  others,  some 
forty  or  fifty  in  number,  the  following  items  of  accu- 
rate description  emerge: 

1.    A  rhombus  is  a  shape  or  figure,  etc. 
It  has  sides  or  lines. 
It  has  four  (sides). 
It  has  straight  (sides). 
It  has  equal  (sides). 

Two  sides  slant  the  same  way  (are  parallel) , 
The  other  two  sides  slant  the  same  way. 
Two  that  slant  the  same  way  are  opposite  each 

other. 

The  other  two  that  slant  the  same  way  are  op- 
posite to  each  other. 
It  has  corners. 
There  are  four  (corners). 
There  are  two  big  (corners). 
And  there  are  two  little  (corners). 
The  two  big  corners  are  opposite  each  other. 
The  two  little  corners  are  opposite  each  other. 
The  two  big  ones  are  equal. 
And  the  two  little  ones  are  equal. 
(A  total  of  17  points.) 


42  INDUCTIVE   VS.   DEDUCTIVE    METHODS. 

2.  A  trapezium  is  a  shape  or  figure,  etc. 
It  has  sides  or  lines. 

It  has  four  (sides). 
It  has  straight  (sides). 
The  sides  are  unequal. 
It  has  corners. 
There  are  four  (corners). 
The  corners  are  unequal. 
(A  total  of  8  points.) 

3.  A  rhomboid  is  a  figure  or  diagram  or  shape. 
It  has  sides  or  lines. 

There  are  four  sides. 

The  sides  are  straight. 

There  are  two  long  sides. 

And  there  are  two  short  sides. 

The  two  long  sides  are  equal. 

And  the  two  short  sides  are  equal. 

The  two  long  sides  are  opposite  each  other. 

And  the  two  short  sides  are  opposite  each 

other. 

The  two  long  sides  slant  the  same  way. 
And  the  two  short  sides  slant  the  same  way. 
It  has  corners. 
There  are  four  (corners). 
There  are  two  big  (corners). 
And  there  are  two  little  (corners). 
The  two  big  corners  are  equal. 
The  two  little  corners  are  equal. 
The  two  big  corners  are  opposite  to  each  other. 
The  two  little  corners  are  opposite  to  each 

other. 

(A  total  of  20  points.) 


FIRST    SERIES   OF   EXPERIMENTS.  43 

4.    A  diagonal  of  a  square  is  a  line. 
It  is  a  straight  line. 

It  is  drawn  from  one  corner  to  another. 
The  corner  to  which  it  is  drawn  is  opposite  the 
other. 

(A  total  of  4  points.) 

If  with  this  scheme  of  marking  in  view  we  turn  to 
Laura  B 's  paper,  we  shall  see  that  she  will  re- 
ceive 5  positive  marks  for  her  definition  of  a  rhom- 
bus; but  she  has  two  'bad  errors/  The  four  sides 
of  the  rhombus  do  not  slant  the  same  way,  and  the 
corners  are  not  the  same  size.  Her  definition  of  a 
trapezium  receives  7  positive  marks;  there  are  no 
negative  marks  for  bad  errors  in  this  definition. 

It  will  now,  doubtless,  be  quite  easy  for  anyone 
with  the  aid  of  the  table  to  assess  the  rest  of  the  posi- 
tive marks.  But  I  might,  perhaps,  call  attention  to 
the  fact  that  there  are  two  more  *  bad  errors '  in  this 
paper.  The  corners  of  a  rhomboid  are  not  all  the 
same  size,  and  a  diagonal  of  a  square  is  not  a  shape. 

Ehoda  T 's  paper  contains  some  errors  in  spell- 
ing, which,  of  course,  are  not  counted  in  experiments 
of  this  kind.  It  can  be  quite  easily  marked  on  the 
system  given  above,  and  I  think  any  teacher  who 
marks  it  will  agree  that  there  are  no  'bad  errors.' 

It  must  not  be  thought  that  children  taught 
inductively  make  no  bad  errors  when  they  apply 
their  knowledge  to  new  material.  A  comparison, 
however,  between  the  number  of  bad  errors  in- 
volved in  the  use  of  the  two  methods  will  be  found 
very  useful  later  on.  All  the  papers  in  all  the  tests 
and  exercises  in  this  school  I  marked  myself,  and  the 


44  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

marks  were  subsequently  checked  by  the  head  mis- 
tress of  the  school. 


10.     Chronology  of  the  Experiment. 

All  the  tests  and  exercises,  with  the  exception  of 
the  Preliminary  Test,  were  taken  on  Tuesday  morn- 
ings at  10.10  A.  M.  All  instructions  and  teaching 
were  given  by  myself,  and  I  was  present  with  the 
children  during  the  whole  time  that  each  exercise  was 
done. 

The  Preliminary  Test  for  the  division  into  two 
equal  groups  was  worked  on  September  25th,  1911. 
The  teaching  and  learning  of  the  first  set  of  defini- 
tions (which  occupied  17  minutes  for  each  group) 
and  an  immediate  test  in  reproduction  were  done  on 
October  3d.  The  second  test  in  reproduction  was 
given  on  October  10th.  The  test  to  see  how  far  the 
children  could  apply  their  knowledge  or  method  to 
new  material  was  given  on  October  17th,  and  the  last 
test  on  the  first  set  of  definitions — a  further  test  in 
deferred  reproduction — was  given  on  November 
7th.  The  lessons  preceding  the  exercises  were,  in 
all  cases,  the  same.  For  writing  out  what  they  knew 
no  time  limit  was  insisted  on :  the  children  were  all 
allowed,  nay  encouraged,  to  put  down  as  much  as 
they  could.  A  note  was  kept  of  the  time  taken  on 
each  occasion.  In  the  test  taken  immediately  after 
teaching  and  learning  25  minutes  was  the  limit,  and 
it  was  noticed  that  the  Deductive  Group,  i.  e.,  those 
who  had  learnt  the  definitions,  were  much  the 
quicker.  The  Inductive  Group  were,  of  course,  to  a 
large  extent,  making  the  definitions  up  as  they  went 
along.  In  the  first  test  of  deferred  reproduction, 


FIKST   SERIES   OF   EXPERIMENTS.  45 

which  took  place  a  week  later,  30  minutes  were  taken ; 
in  the  test  to  show  the  power  of  application  to  new 
material,  32  minutes;  and  in  the  second  test  of  de- 
ferred reproduction,  which  took  place  a  month  after 
the  acquisition  of  the  original  definitions,  32  minutes 
were  taken.  It  is  possible  to  ascribe  this  lengthening 
of  the  time  of  the  exercise  to  an  increasing  difficulty 
of  remembrance.  This  may  be  a  factor,  but  I  am  in- 
clined to  think  there  may  be  another;  the  children 
may  be  getting  more  thoughtful  over  the  work,  and 
consequently  slower. 

11.    Results, 
(a)     The  Marks  for  the  Preliminary  Tests. 

Every  mark  which  every  child  obtained  in  every 
exercise  was  carefully  tabulated,  though  from  the 
final  table  it  was  necessary  to  exclude  three  or  four 
children  who  had  been  absent  on  several  occasions. 
The  first  and  last  child  in  the  Inductive  Group  were 
among  these  cases,  so  the  corresponding  children, 
namely,  the  first  and  last  of  the  Deductive  Group, 
were  also  omitted.  There  were  then  21  cases  left  in 
each  group. 

In  the  Preliminary  Test,  in  which  the  children 
tried  by  themselves  to  see  what  they  could  do  in  de- 
fining square,  triangle,  etc.,  the  group  which  subse- 
quently did  the  inductive  work  gained  an  average 
mark  of  9.4,  with  a  mean  variation  of  2.2;  and  the 
group  which  subsequently  learnt  the  definitions  de- 
ductively obtained  an  average  mark  of  9.5,  with  a 
mean  variation  of  2.3.  The  highest  mark  in  each 
group  was  15 ;  the  lowest  mark  was  6.  In  so  far  as 
it  is  possible  to  make  a  satisfactory  division  of  a 
class  on  one  test  only  the  groups  were  well  balanced. 


46  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

(b)     The  Marks  for  the  Test  Immediately  After  the 
Definitions  Had  Been  Taught  and  Learnt. 

The  Inductive  Group  gained  an  average  mark  of 
22.6  of  positive  marks,  with  a  mean  variation  of  2.5, 
whilst  the  Deductive  Group  gained  an  average  mark 
of  25.6,  with  a  mean  variation  of  2.2.*  This  is  a 
clear  and  significant  difference  in  favor  of  the  De- 
ductive Group — a  difference  which  is  accentuated 
when  'bad  errors'  are  taken  into  account,  for  the 
former  group  makes  21  bad  errors  and  the  latter 
group  only  12.  Deducting  the  negative  marks — the 
marks  for  bad  errors — from  the  positive  marks,  it  is 
found  that  the  Inductive  Group  scores  an  average  of 
21.6  marks  (mean  variation  2.6),  and  the  Deductive 
Group  scores  an  average  of  25.0  (mean  variation 
2.6) — a  still  clearer  and  more  significant  difference. 
It  is  highly  probable  from  the  above  average  marks 
and  variabilities  that  this  difference  is  a  difference, 
so  to  speak,  all  along  the  line,  i.  e.,  one  which  will  be 
found  between  both  the  best  pupils  of  each  group 
and  also  between  the  worst.  That  relationship,  how- 
ever, will  be  shown  more  clearly  by  the  following 
table : 

Table  I,  showing  the  work  of  the  Inductive  and  Deductive  Groups 
compared,  in  the  Preliminary  Test,  and  in  the  First  Test  after 
the  definitions  had  oeen  learnt  and  taught  (positive  marks 
only). 

Group  A  ( Inductive) .        Group  B  ( Deductive) . 

Av.  mark 

Marks  in  No.          in       Av.  mark    No.   Av.  mark  Av.  mark 

preliminary  of       prelim,     in  first       of    in  prelim,    in  first 

tests.  girls.       test,    final  test,  girls.       test,     final  test. 

12  and  over 5          13.0          25.4          5          13.4          25.4 

9,   10,   11 7          10.0          22.0          7          10.0          25.1 

7,    8 5  7.8          21.4          5  7.6          26.6 

6 4  6.0          21.7          4  6.0          25.5 


*The  difference  between  the  averages  is  about  five  times  the 
probable  error  of  their  difference,  even  on  the  assumption  that  the 
series  are  not  positively  correlated. 


FIRST    SERIES   OF   EXPERIMENTS.  47 

An  inspection  of  Table  I  shows  that,  while  there 
is  no  difference  in  the  results  between  the  two  sec- 
tions of  the  ablest  children,  those  at  the  top  of  each 
group,  there  are  considerable  differences  in  the  re- 
maining sections.  The  argument  of  the  teacher 
whose  letter  I  have  given,  namely,  that  routine  meth- 
ods are  better  for  the  immediate  reproduction  of 
the  actual  material  learnt,  must  be  conceded  for  chil- 
dren of  this  level  of  ability.  A  teacher  once  asked 
me  rather  scornfully:  "What  did  you  expect  from 
your  17  minutes'  teaching?"  Not  much,  perhaps, 
but  we  shall  see  more  about  that  later.  And,  of 
course,  there  were  also  17  minutes'  learning,  so  the 
comparison  was  a  fair  one  in  any  case. 

(c)    The  Marks  for  Tests  of  Deferred  Reproduction. 

It  is,  however,  one  thing  to  answer  questions  im- 
mediately after  one  has  learnt,  or  has  been  taught, 
the  answers;  perhaps  it  is  quite  another  thing  to 
give  those  answers  accurately  by  and  by. 

The  second  Final  Test  took  place  one  week  after 
the  first  Final  Test,  and  the  children,  as  I  have 
already  pointed  out,  did  not  know  they  were  ever  to 
have  the  exercises  again.  It  is  a  matter  of  great 
importance  to  the  teacher,  and  it  is  a  matter  of 
great  importance  to  the  experimenter  to  know  how 
far  the  immediate  results  from  the  work  of  any 
group  of  children  may  be  taken  as  fairly  represent- 
ing what  that  group  of  children  will  do  later  on. 
And,  perhaps,  a  week  is  too  short  a  time.  "The 
children  won't  have  forgotten  all  about  it  by  then," 
as  a  teacher  said,  so  a  third  Final  Test — the  second 
test  of  deferred  reproduction — was  given  a  month 


48  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

later.    Let  me  present  a  few  comparisons.    I  will 
give  first  the  positive  marks  only : 

Table  II,  showing  the  relation  betiveen  immediate  and  deferred 
reproduction  (positive  marks  only). 

Prelinri-       First        Second       Third 


Inductive  Group. 

Average.  , 
M.  V  

nary 
test 
.  .   9.4 
22 

final 
test. 
22.6 
2.5 

final 
test. 
23.2 
2.0 

final 
test. 
22.0 
3.1 

Deductive  Group. 

Average.  . 
M.  V. 

.  .    9.5 
.    2.3 

25.6 
2.2 

25'.4 
2.1 

24*.9 
1.9 

Inductive  Group. 

Average. 
M.  V 

nary 
test. 
...   9.4 
2.2 

final 
test. 
21.6 
2.6 

final 
test. 
22.2 
2.2 

Deductive  Group. 

Average. 
M.  V. 

.  .  .    9.5 
2.3 

25.0 
2.6 

24.9 
2.6 

I  will  next  show  the  immediate  and  deferred  re- 
sults for  the  two  groups,  when  the  negative  marks 
for  the  Final  Tests  have  been  subtracted  from  the 
positive  marks : 

Table  III,  showing  the  relation  between  immediate  and  deferred 
reproduction. 

Prelimi-      First         Second        Third 

final 
test. 
21.0 

3.0 
24.1 

2.1 

The  results  from  both  the  tables  are  in  marked 
agreement.  Both  groups  of  children  have  gone  down 
somewhat,  but  the  children  taught  inductively  have 
lost  less  of  their  original  knowledge  than  the  group 
which  worked  deductively.  It  appears  that  for 
groups  of  children  of  this  age  and  ability  the  imme- 
diate results  of  these  methods  of  learning  and  teach- 
ing may  be  accepted  as  indicating,  comparatively, 
not  only  what  the  children  can  do  at  the  time,  but  also 
what  the  groups  will  do  by  and  by. 

This  may  be  shown  roughly  in  the  following  table : 


FIRST   SERIES   OF   EXPERIMENTS.  49 

(d)     Correlation  Between  Immediate  and  Deferred 
Reproduction. 

Table  IV,  showing  the  correlation  bettveen  immediate  and  deferred 
reproduction  in  the  two  groups,  section  l)y  section  (positive 
marks  only). 

Group  A  (Inductive).       Group  B  (Deductive). 

Av.        Av.       Av.  Av.       Av.       Av. 

mark    mark    mark  mark   mark   mark 

Marks  in          No.    first    second  third    No.    first   second  third 

preliminary  of    final      final     final     of     final     final     final 

test.  girls,    test.     test.     test,  girls,  test.     test.     test. 

12  and  over 5      25.4      25.6      25.2       5       25.4      24.8      25.0 

9,   10,    11 7       22.0       24.0       23.4       7       25.1       24.4       23.6 

7,  8 5      21.4      20.8       19.0      5       26.6      26.2       25.2 

6 4      21.7      21.7       19.0      4      25.5      27.0       26.2 

It  is  fairly  obvious  that  positive  correlation  exists 
between  the  children's  immediate  work,  their  work 
after  one  week's  interval,  and  their  work  after  a 
month's  interval.  But  there  is  some  irregularity 
here  and  there,  so  that  it  will  be  better  to  set  down 
each  child's  individual  results  and  work  out  the  cor- 
relations between  them  rather  than  trust  wholly  to 
the  inspection  of  the  averages  of  these  corresponding 
sections.  The  lists  were  arranged  thus : 

Inductive  Group  (positive  marks  only). 

Name.                  First  final  Second  final  test.  Third  final  test. 

( Initials  only. )               test.  A  week  later.      A  month  later. 

E.  S 29  24                            26 

M.  J 27  27                            22 

D.  S 21  24                            23 

E.  B 25  25                            27 


E.    W 20  20  17 

(21  cases.) 

A  similar  table  was  made  of  the  individual  results 
from  all  the  children  who  worked  in  the  Deductive 
Group,  which  also  contained  21  cases.  Then  the 
Pearson  coefficient  of  correlation,  or  *r'  formula, 


50  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

Sxy 

which  runs  thus  :  r  =  -     -  ,  was  applied  to  the  in- 

7 


dividual  cases,  and  the  following  results  were  ob- 
tained : 

Inductive  Deductive 

group.  group. 
Correlation  between  results  of  first  and 

second  final  tests  ....................       +  .537  +  .532 

Correlation  between  results  of  first  and 

third  final  tests  ......................       +  .589  +  .331 

i 

It  certainly  appears  that  whilst  the  averages  for 
the  groups  have  remained  remarkably  steady,  the 
coefficients  of  correlation  show  that  the  individual 
children  have  changed  places  considerably.  But  we 
must,  I  think,  admit  that  an  immediate  test  of  the 
result  of  a  method  of  teaching  or  learning  is  one 
which  gives  us  reasonable  ground  to  expect,  from  the 
group  as  a  whole,  a  similar  result  later  on.  That  is, 
for  reproductive  exercises,  where  fairly  homogene- 
ous groups  of  school  children  are  concerned,  tests  of 
immediate  reproduction  and  tests  of  deferred  repro- 
duction give  very  similar  results.  We  must  appar- 
ently concede  the  point  argued  for  in  the  head  mas- 
ter's letter  previously  quoted,  viz.,  that  a  mechan- 
ical method  is  better  either  for  immediate  or  de- 
ferred reproduction  at  examinations.  Let  us  concede 
that  point,  bearing  in  mind  that  our  examination 
results,  so  far,  have  been  only  of  a  kind  in  which  the 
exact  reproduction  has  been  asked  for  of  precisely 
what  was  taught  or  learnt. 

(e)     Results  When  the  Two  Groups  Are  Tested  on 
New  Material. 

We  have  seen,  hitherto,  that  the  Deductive  Group 
has  scored  higher  marks  than  the  Inductive  on  every 


FIKST   SERIES   OF   EXPERIMENTS.  51 

occasion.  Certainly  our  preliminary  division  of  the 
two  groups  on  one  test  only  is  open  to  criticism,  and 
we  might  suppose  that  we  had  not  really  succeeded 
in  obtaining  equal  groups.  And  possibly  we  have 
not,  though  I  think  it  likely  that  they  were  approxi- 
mately equal  in  initial  capacity.  Whether  that  be 
the  case  or  not,  there  is  found  a  definite  difference 
between  the  two  groups  on  the  results  of  a  test  on  new 
material,  and  in  the  opposite  direction  from  the  pre- 
vious difference.  The  group  taught  inductively  now 
leads  the  way.  Counting  only  positive  marks,  the 
difference  is  small:  the  Inductive  Group  scores  an 
average  mark  of  20.9  (mean  variation  3.8),  whilst 
the  Deductive  Group  scores  an  average  mark  of  20.0 
(mean  variation  2.7).  But  when  the  marks  for  'bad 
errors '  are  subtracted  there  is  a  distinct  and  decided 
difference:  the  Inductive  Group  scores  an  average 
mark  of  19.4  (mean  variation  3.5),  whilst  the  Deduc- 
tive Group  scores  an  average  mark  of  17.7  (mean 
variation  2.5).*  The  more  mechanical  method  has 
produced  a  much  larger  crop  of  'bad  errors/  for 
there  is  an  average  of  2.3  per  child  in  this  group, 
against  an  average  of  1.2  in  the  other : 

Table  V,  showing  the  'bad  errors'  of  the  two  groups. 

Inductive  Deductive 

group.  group. 

No.  of  girls  with  5  bad  errors 0  2 

No.  of  girls  with  4  bad  errors 0  3 

No.  of  girls  with  3  bad  errors 3  3 

No.  of  girls  with  2  bad  errors 5  7 

No.  of  girls  with  1  bad  error 6  4 

No.  of  girls  with  0  bad  errors 7  2 


*The  difference  between  the  means  is  more  than  twice  its  prob- 
able error,  even  on  the  assumption  that  the  series  are  uncorrelated. 


52  INDUCTIVE   VS.    DEDUCTIVE    METHODS. 

For  children  of  this  age  and  capacity,  therefore, 
we  are  entitled  to  urge  that  the  inductive  method  is 
much  less  provocative  of  error  when  'new  material ' 
is  given  for  test  purposes.  On  the  old  material  the 
Inductive  Group  made  more  errors  than  the  Deduc- 
tive in  every  test,  so  that  we  cannot  suppose  that  the 
result  with  new  material  is  consequent  upon  a 
greater  initial  tendency  to  '  howlers'  in  the  De- 
ductive Group ;  indeed,  the  tendency  seems  the  other 
way,  if  there  be  one.  We  are  often  warned  that 
averages  are  prone  to  conceal  important  differences 
between  individuals,  but  I  cannot  expect  huge  tables 
of  individual  results  to  be  printed,  and,  indeed,  no 
useful  conclusions  in  experiments  of  this  kind  could 
be  drawn  from  individual  results  as  such  if  they  were 
printed ;  they  must  be  grouped  if  they  are  to  be  of 
service.  But  I  can  show  how  far  the  superiority 
holds  or  fails  for  the  sections  of  corresponding  ini- 
tial ability  in  the  two  groups : 

Table  VI,  showing  the  work  of  the  two  groups  compared,  section 
by  section,  in  the  last  reproductive  test,  and  in  the  test  of 
application  to  new  material,  the  marks  for  bad  errors  being 
subtracted  from  the  positive  marks. 

Group  A  (Inductive).         Group  B  (Deductive). 

Av. 

Av.  mark  Av.  mark  Av.  mark  mark  ap- 

Marks  in         No.          last     application    No.          last       plication 
preliminary        of      reproduc-    to  new        of      reproduc-    to  new 

test.  girls,     tivetest.  material,    girls,    tivetest  material. 

12  and  over..  5  23.8  20.4  5  24.4  17.6 

9,  10,  11 7  22.8  21.7  7  22.9  17.0 

7,8 5  18.6  17.4  5  25.2  19.0 

6 4  18.2  18.2  4  25.7  18.5 

In  the  case  of  every  corresponding  section  the  loss, 
when  application  is  made  to  new  material,  is  greater 
in  the  Deductive  Group  than  in  the  Inductive  Group. 


FIRST   SERIES   OF   EXPERIMENTS.  53 

12.    Pedagogical  Conclusions. 

I  have  previously  pointed  out  that,  as  a  whole,  the 
Inductive  Group  gains  higher  marks  for  application 
to  new  material  than  the  Deductive.  What  pedagog- 
ical conclusions  may  we  draw  from  this?  Two  out- 
standing conclusions  seem  to  me  to  follow  from  this 
work:  the  first  relating  to  teaching,  the  second  re- 
lating to  examination. 

First,  as  to  the  method  in  teaching:  I  suppose  no 
teacher  would  desire  us  nowadays  to  favor  a  method 
merely  because  it  enabled  us  to  produce  a  better  re- 
sult in  the  exact  reproduction  of  what  had  been 
taught  or  learnt.  Let  us  consider  exact  reproduction 
as  so  much  to  the  good,  but  let  us  also  remember  that 
individual  lessons  form,  or  should  form,  part  of  a 
course,  and  the  method  which  enables  a  pupil  to  make 
the  best  attack  on  new  analogous  material  is,  one 
may  reasonably  suppose,  likely  to  emerge  triumph- 
ant at  the  end  of  the  course.  Such  a  method  does 
really  train  *  intelligence,'  in  the  best  sense  of  that 
much-abused  word. 

Secondly,  as  to  method  in  examination.  Whenever 
examiners  set  work  to  be  done  which  is  a  mere  repro- 
duction of  what  the  children  have  been  taught  or 
have  learnt,  they  are  favoring  the  mechanical 
method  rather  than  the  inductive  one. 

I  do  not  say  there  is  no  place  for  examination  of 
that  sort,  but  high  assessments  for  teaching  should 
never  be  given  on  such  a  basis.  The  supreme  test  of 
good  teaching  is  the  power,  on  the  part  of  the  pupils, 
to  attack  'new  material.'  One  word,  however,  of 
caution.  I  do  not  mean  material  wholly  new,  as  many 
psychological  tests  are.  By  the  use  of  such  tests  as 


54  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

those  we  are  measuring  natural  ability  rather  than 
the  result  of  pedagogical  work.  The  material  should 
be  'new,'  but  it  should  be  analogous  to  the  work 
which  the  pupil  may  reasonably  be  expected  to  have 
done  before. 

I  venture  to  suggest  that  examinations  of  this  kind 
would  raise  the  tone  and  method  of  teaching  rather 
than,  as  too  often  has  been  and  is  the  case,  tend 
to  depress  them.  I  wish  to  exempt  junior  scholar- 
ship examinations.  They  should,  in  my  judgment,  be 
psychological  in  the  sense  given  above. 

But,  perhaps,  the  reader  may  feel  that  I  am  build- 
ing up  a  huge  structure  of  theory  on  the  basis  of  a 
very  little  experiment ;  so  I  will  turn  to  the  second 
series  of  experiments  in  this  research  and  show  how 
far  the  facts  and  conclusions  resemble  those  of  the 
first. 


V.    SECOND  SERIES  OF  EXPERIMENTS. 
1.    General  Plan. 

/ 

As  before,  a  whole  class  of  children,  under  the 
same  teacher,  studying  the  same  curriculum  in  ac- 
cordance with  the  same  time-table  of  school  work, 
was  divided  into  two  equal  groups  on  the  basis  of 
preliminary  tests  in  geometrical  definition,  which  the 
children  attempted,  untaught  and  unaided.  Then 
the  pupils  of  one  group  were  taught  inductively  how 
to  arrive  at  the  definitions,  whilst  the  other  group 
learnt  the  definitions  deductively.  An  immediate 
test  was  given  to  show  which  method  was  the  better 
for  the  purposes  of  immediate  reproduction,  and  sub- 
sequent tests  were  given  on  the  same  subject-matter 
to  indicate  which  of  the  two  groups  was  the  more 
successful  in  deferred  reproduction.  Also,  as  before, 
a  test  was  given  on  new  analogous  material  to  see 
which  of  the  two  methods  showed  the  greater  *  trans- 
fer7 effect.  One  or  two  outstanding  points  of  differ- 
ence between  the  conditions  of  this  experiment  and 
those  of  the  first  experiment  may,  perhaps,  usefully 
be  mentioned  here  before  proceeding  to  the  details. 
The  children  who  did  the  work  were,  as  before,  girls 
belonging  to  an  elementary  school  in  London.  But 
in  this  case  they  were  of  a  poorer  social  class ;  they 
were  older  than  the  girls  in  the  previous  school,  their 
average  age  amounting  to  rather  more  than  13  years, 

55 


,56  INDUCTIVE   VS.    DEDUCTIVE   METHODS. 

and,  measured  by  school  standards,  they  were  more 
proficient  mentally,  for  these  children  were  graded 
as  Standard  VI,  a,  and  VII,  whereas,  in  the  previous 
school,  the  children  were  graded  as  Standard  V.  The 
children  of  this  class  had  done  a  great  deal  of  manual 
constructive  work,  and  were  taught  by  a  teacher 
from  whom  they  had  learnt  to  express  their  meaning 
in  direct,  simple  language.  Like  the  girls  of  the  pre- 
vious school,  they  knew  nothing  of  geometrical  defi- 
nition; and  '  Demonstrative  Geometry,'  or  even  *  Eu- 
clid, '  were  terms  of  no  meaning  either  to  themselves 
or  to  their  parents. 

There  was,  too,  an  important  difference  in  the 
early  part  of  the  procedure  of  this  experiment  from 
that  of  the  previous  one.  Instead  of  dividing  the  chil- 
dren on  the  basis  of  one  test,  four  tests  were  given, 
in  which  the  same  material  was  employed  through- 
out. It  was  believed  that  the  division  into  equal 
groups  would  be  much  more  satisfactory  if  it  were 
effected  on  a  wider  basis  than  the  results  of  one  test ; 
and,  indeed,  the  greater  regularity  of  the  subsequent 
work  showed  that  to  be  the  case.  Thirty-seven  chil- 
dren started  the  experiment,  but  unavoidable  ab- 
senses  from  school  reduced  the  number  available  for 
the  tabulation  in  *  equal  groups '  to  34. 

2.     The  Preliminary  Tests  and  the  Method  of 
Marking. 

As  in  the  previous  case,  drawings  of  squares,  tri- 
angles, oblongs,  and  diameters  of  circles,  with  their 
names  appended,  were  shown  to  the  children,  and  the 
questions,  "What  is  a  square? "  etc.,  were  written 
on  the  blackboard.  The  units  of  marking,  as  before, 


SECOND   SERIES    OF   EXPERIMENTS.  57 

were  obtained  from  a  careful  study  of  the  answers 
actually  given  by  the  children. 

One  or  two  instances  of  the  children's  spontaneous 
attempts  at  definition  may  possess  psychological  in- 
terest. 

Lily  H ,  aged  13  years  6  months,  a  girl  graded 

as  Standard  VI,  a,  wrote : 

1.  A  square  is  four  lines  each  of  the  same  length  all  joining 
one  another  and  when  they  are  joined  they  form  a  square  with 
four  angles  the  square  may  be  straight  up  or  slanting. 

2.  A  triangle  is  a  three  line  drawing,  joining  each  at  the  ends 
and  when  it  is  drawn  it  forms  a  drawing  with  three  angles  each 
of  the  same  size. 

3.  An  oblong  is  a  figure  with  four  lines  same  as  the  square  only 
there  are  two  long  lines  and  two  of  them  are  short  lines  with  four 
angles  of  the  same  size. 

4.  A  diameter  is  the  line  drawn  through  a  circle  to  separate 
one  half  from  the  other  only  it  must  be  drawn  through  the  centre. 

/ 

The  above  is  one  of  the  better  papers  which  were 
worked  during  the  Preliminary  Tests,  but  it  is  cer- 
tainly not  the  best.  Let  me  now  give  an  inferior  one. 

Ada  B ,  aged  13  years  1  month,  also  graded  as 

Standard  VI,  a,  wrote : 

1.  A  square  is  a  thing  with  four  equal  sides.    A  square  can  be 
all  different  shapes  as  long  as  the  four  sides  are  equal. 

2.  A  triangle  is  something  which  has  three  sides  and  the  sides 
must  be  as  long  as  each  other. 

3.  An  oblong  has  four  sides,  two  of  the  lines  are  short  and  two 
are  long.    The  two  long  lines  must  face  each  other,  and  the  short 
ones  must  be  the  same  length  as  each  other. 

4.  A  diameter  of  a  circle  is  a  round  ring  divided  into  half. 

It  was  seen,  after  a  careful  perusal  of  the  chil- 
dren's papers,  that  the  units  of  marking  which  had 
been  worked  out  for  use  in  the  previous  school  were 
also  quite  suitable  for  this  one.  Turning  to  the  an- 
swers of  Lily  H.,  and  marking  with  these  units,  we 
find  that  in  her  first  answer  she  has  a  mark  for 


58  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

'lines,'  another  for  'four'  (lines),  a  third  for  'of  the 
same  length,'  a  fourth  for  'angles,'  and  a  fifth  for 
'four'  (angles).  The  lines  do  not  exactly  all  join  one 
another,  but  the  statement  was  not  considered  an 
error ;  and  the  further  statement, '  the  square  may  be 
straight  up  or  slanting'  was  considered  irrelevant. 
In  the  second  answer  Lily  H.  receives  a  mark  for 
'drawing,'  one  for  'lines,'  one  for  'three'  (lines),  one 
for  'angles,'  and  one  for  'three'  (angles) ;  a  total  of 
five  marks.  She  describes  the  angles  as  being  all  of 
the  same  size,  which  is  certainly  a  serious  error,  and 
was  probably  due  to  the  confinement  of  her  attention 
to  the  equilateral  triangle,  which  was  one  of  the  tri- 
angular figures  shown.  For  her  definition  of  oblong 
she  receives  a  mark  for  'figure,'  another  for  'lines,' 
a  third  for  'four'  (lines),  a  fourth  for  'two  long 
lines,'  a  fifth  for  'two  short  lines,'  and  a  sixth  and 
seventh  for  'angles'  and  'four'  (angles).  For  her 
fourth  definition,  namely,  that  of  a  diameter  of  a 
circle,  she  receives  a  mark  for  'line,'  and  another  for 
'drawn  through  the  center.'  Lily  H.  thus  receives  a 
total  of  19  positive  marks. 

Ada  B 's  paper,  marked  in  the  same  way,  re- 
ceives a  total  of  11  positive  marks.  Her  third  an- 
swer— the  definition  of  an  oblong — is  quite  unex- 
pectedly good,  considering  the  weakness  of  her  defi- 
nitions of  'triangle'  and  'diameter.'  With  various 
triangles,  mostly  scalene,  exposed  before  her  eyes,  it 
was  certainly  a  'bad  error'  to  say  "the  sides  must  be 
as  long  as  each  other. ' '  Probably,  with  these  explan- 
ations, the  method  of  marking  adopted  will  be  read- 
ily applicable.  I  will  now  set  out  the  chronology  of 
the  whole  experiment. 


SECOND   SEEIES   OF   EXPERIMENTS.  59 

3.    Chronology  of  the  Experiment. 

First  of  all,  a  preparatory  exercise  was  given  at 
9.40  A.  M.  on  Friday,  October  20th,  1911,  under  test 
conditions,  to  accustom  the  children  to  work  of  this 
kind,  which  was  quite  new  to  them.  Then,  on  Tues- 
day, October  24th,  Wednesday,  October  25th,  and 
Friday,  October  27th,  at  9.40  in  the  morning,  imme- 
diately after  Scripture  lesson,  a  second,  third  and 
fourth  Preliminary  Test  were  given.  On  the  results 
of  the  second,  third  and  fourth  tests  the  class  was 
divided  into  two  equal  groups.  Then  on  Friday,  No- 
vember 3d,  at  the  same  time  in  the  morning,  the 
pupils  of  Group  B  were  taught  the  definitions  in- 
ductively by  me  in  the  way  previously  explained, 
whilst  Group  A  learnt  the  definitions  by  studying 
them  as  written  out,  and  referring  to  the  illustrative 
figures  drawn  beneath  the  written  definitions.  I  took 
care  that  the  children  who  were  learning  the  defini- 
tions should  receive  all  their  instructions  from  me, 
and  informed  the  children  of  both  groups  that  they 
would  be  required  to  answer  questions  immediately 
afterwards  about  what  they  were  learning  or  being 
taught,  respectively.  The  children  who  had  been  try- 
ing for  themselves  without  help  for  four  exercises  to 
see  what  they  could  do  in  the  way  of  spontaneous 
definition  were,  of  course,  in  a  state  of  receptivity 
for  instruction  of  either  kind,  inductive  or  deductive. 
Their  marks  had  risen  steadily  day  by  day,  so  that 
they  were  still  in  the  *  improving'  stage  for  this  work. 
The  teaching  lasted  17  minutes,  and,  of  course,  the 
same  time  was  allotted  to  the  study  of  the  written 
definitions.  The  two  groups  were  put  together  imme- 
diately, and  the  old  questions :  " What  is  a  square?" 


60  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

etc.,  were  written  on  the  blackboard.  It  was  noticed 
that  the  children  working  in  the  deductive  group  had 
answered  their  questions  some  three  to  five  minutes 
sooner  than  the  girls  in  the  inductive  group. 

Four  days  later,  on  Tuesday,  November  7th,  a  sec- 
ond test  of  precisely  similar  nature  was  given  to 
both  groups  at  9.40  A.  M.,  after  Scripture  lesson,  as 
before. 

On  Friday,  November  10th,  at  the  same  time  in  the 
morning,  and  after  the  same  lesson  as  on  previous 
occasions,  a  test  was  given  to  both  groups  on  new 
analogous  matter  to  test  the  comparative  '  transfer' 
values  of  the  two  methods  of  learning. 

The  last  test  in  the  series  was  given  at  the  same 
time  in  the  morning,  and  after  Scripture,  as  before, 
on  Friday,  December  1st.  In  this  test  the  previous 
questions  on  the  material  actually  studied  were  given 
again — "What  is  a  square  1"  etc.  The  object  of  this 
test — the  second  test  in  deferred  reproduction — was 
to  discover,  if  possible,  which  of  the  two  groups  had 
lost  the  more  after  a  considerable  interval ;  in  other 
words,  which  method  of  teaching  or  learning  favored 
the  more  permanent  retention.  I  ought  to  say  that, 
with  the  exception  of  the  test  given  immediately 
after  the  teaching  and  learning,  the  children  were 
not  aware  that  they  were  going  to  do  any  of  these 
tests  before  they  were  actually  set  to  do  them. 

4.     The  Final  Tests  and  the  Method  of  Marking. 

Three  of  the  Final  Tests  were  repetitions  of  the 
Preliminary  Tests,  and  the  same  method  of  marking 
was  adopted  in  them  as  in  the  Preliminary  Tests. 
These  were  the  tests  given  immediately  after  teach- 


SECOND   SEEIES    OF    EXPERIMENTS.  61 

ing  and  learning  and  the  two  tests  of  deferred  repro- 
duction. The  remaining  test,  in  which  the  children, 
without  further  teaching,  were  required  to  attack 
new  material,  was  identical  with  the  corresponding 
test  given  in  the  previous  school,  and  it  was  marked 
in  the  same  way.  Drawings  of  rhombuses,  trapezi- 
ums, rhomboids  and  diagonals  of  squares,  with  their 
names  appended,  were  shown  to  the  children,  and 
they  were  required  to  answer  in  writing  the  ques- 
tions :  ' '  What  is  a  rhombus  I ' '  etc. 

5.    Results  of  the  Experiments, 
(a)     Results  of  the  Preliminary  Tests. 

The  marks  obtained  in  the  Preparatory  Test  will 
not  be  given ;  it  was  noticed  that  though  the  children 
at  the  top  and  bottom  of  the  lists  remained  much  the 
same,  a  considerable  number  of  children  changed 
places  from  the  Preparatory  to  the  first  Preliminary 
Test.  As  it  was  very  important  that  the  work  of  the 
children  should  be  ' steady'  before  the  class  was  di- 
vided into  two  equal  groups,  two  more  tests — the 
second  and  third  Preliminary — were  given  and  the 
results  correlated.  The  work  started  with  37  chil- 
dren, but  two  had  been  absent  during  the  tests,  so 
they  were  excluded  from  the  lists. 

The  results  of  the  three  Preliminary  Tests  are 
shown  compendiously  in  the  following  table : 

Table  VII,  showing  the  correlation  between  the  results  of  the  First, 

Second  and  Third  Preliminary  Tests. 

Marks  in  first  No.  of      Av'age  Marks  in  Preliminary  Tests, 

preliminary  test.  children.    First  test.    Second  test.    Third  test. 

19,  18 7  18.3  18.7  19.0 

17,  16 5  16.4  18.4  17.6 

15,  14 9  14.7  16.4  17.0 

13,  12 7  12.7  13.9  14.7 

11  to  6 7  9,6  10.1  12.3 


62  INDUCTIVE   VS.    DEDUCTIVE   METHODS. 

It  is  obvious  from  Table  VII  that  high  positive 
correlation  exists  between  the  results  of  the  first, 
second  and  third  Preliminary  Tests,  and  that  we  are 
measuring  a  mental  function,  or  group  of  mental 
functions,  which  is  working  very  steadily.  A  precise 
numerical  value  for  the  coefficient  of  correlation  has 
been  worked  out  from  the  individual  cases  by  means 

2xy 

of  the  Pearson  formula  r  = ,  and  'r'  has  been 

noi02 

found  to  be  +  .80,  with  a  probable  error  of  .04,  for 
Tests  1  and  2,  and  +  .77  (probable  error  .05)  for 
Tests  2  and  3.  It  seems  very  likely  that  a  division 
of  the  class  into  two  equal  groups  on  the  basis  of 
such  regular  results  as  these  will  be  satisfactorily 
effected. 

The  children  were  divided  into  two  groups  of  17 
girls  each,  thus  (N.  G.,  the  girl  at  the  bottom  of  the 
list,  was  omitted) : 

Table  VIII,  showing  the  division  into  two  equal  groups. 

Group  A. 
r- Marks  for  Preliminary  Tests.— > 


Name. 
(Initials  only.) 
W.    F  

First 
test. 
18 

Second 
test. 
24 

Third 
test. 
21 

Total 
marks. 
63 

H.  L  

18 

20 

19 

57 

H.   G  

17 

20 

17 

54 

G.  B  

11 

11 

11 

33 

T    F  

8 

9 

13 

30 

Averages  

14  2 

15.8 

163 

463 

M.   V.'s.. 

2.3 

3.0 

2.6 

SECOND    SERIES    OF    EXPERIMENTS.  63 

Group  B. 
r-Marks  for  Preliminary  Tests.^ 


Name. 
(Initials  only.) 
D.  A  

First 
test. 
17 

Second 
test. 
22 

Third 
test. 
22 

Total 
marks. 
61 

W.  E  

19 

20 

19 

58 

H    L      

18 

17 

20 

55 

A.   R  

10 

10 

11 

31 

L.   L  

11 

7 

13 

31 

Averages  

14.5 

15.5 

16.3 

46.3 

M.   V.'s.. 

2.6 

2.8 

2.2 

Care  was  taken  also  that  the  children  should  be  so 
arranged  in  the  grouping  that  the  ages  of  the  one 
group  should  very  closely  approximate  to  those  of 
the  other.  The  average  age  of  Group  A  worked  out 
to  13  years  1  month  (mean  variation  7.2  months), 
and  of  Group  B  to  13  years  0.5  months  (mean  varia- 
tion 5.6  months). 


(b)    Results  of  the  Tests  in  Immediate  and  Deferred 
Reproduction. 

It  now  remains  to  be  shown  what  marks  were  ob- 
tained after  the  one  group  had  been  taught  the  defi- 
nitions and  the  other  group  had  learnt  them.  The 
total  marks  will  be  shown  for  the  three  Preliminary 
Tests,  with  the  marks  for  immediate  reproduction 
and  for  the  two  tests  of  deferred  reproduction — the 
one  given  a  few  days  later  and  the  other  about  a 
month  later  than  the  test  of  immediate  reproduction : 


64 


INDUCTIVE   VS.   DEDUCTIVE   METHODS. 


Table  IX,  showing  the  work  of  the  Inductive  and  Deductive 
Groups  compared,  section  by  section,  in  the  Preliminary  Tests 
and  in  the  Tests  of  Reproduction  (positive  marks  only). 


Group  A  (Deductive). 


Marks  for  , Average  Marks -^ 

three                  No.            Pre-             First  Second  Third 

preliminary             of  liminary        repro-  repro-  repro- 

tests.             children,  tests.  duction.  duction,  ductioii. 

Over  50 6              18.4              26.8              26.5  26.0 

40  to  50 6               15.6               27.7               26.5  25.0 

30  to  40 5              11.7              25.0              22.6  23.0 


Group  B  (Inductive). 


Marks  for  , Average  Marks ^ 

three                  No.            Pre-             First  Second  Third 

preliminary             of  liminary  repro-          repro-  repro- 

tests.             children,  tests.  duction.  duction.  duction. 

Over  50 6              18.4              28.8              28.3  26.8 

40  to  50 6               15.4               26.5               26.2  25.2 

30  to  40 5              11.7              25.8              24.8  25.2 


It  seems  clear  that  in  this  case  the  children  taught 
inductively  were  just  as  successful  as  those  taught 
deductively,  even  in  immediate  reproduction,  and 
that  after  a  month's  interval  they  were  rather  more 
so ;  they  had  lost  less  of  what  they  had  been  taught. 
This  will,  perhaps,  be  shown  more  clearly  in  the  fol- 
lowing tables : 


Table  X,  showing  the  work  of  the  two  groups  compared,  in  the 
Preliminary  Tests,  and  in  the  Tests  of  Immediate  and  Deferred 
Reproduction  (positive  marks  only). 


-Average  Marks.- 


Average  mark  First 

for  three  pre-  repro- 

liminary  tests,  duction. 

Inductive  group 15.4  27.1 

M.  V.'s 2.4  1.8 

Deductive  group 15.4  26.6 

M.  V.'s 2.4  1.9 


Second          Third 
repro-  repro- 

duction,       duction. 


26.5 
1.9 

25.4 
2.2 


25.8 
2.2 

24.8 
2.4 


SECOND   SEKIES   OF    EXPERIMENTS.  65 

Table  XI,  showing  the  work  of  the  tivo  groups  compared  in  the 
Preliminary  Tests  and  in  the  Tests  of  Immediate  and  Deferred 
Reproduction  (when  the  negative  marks  have  been  subtracted 
from  the  positive  marks). 

/ Average  Marks.- 

Average  mark  First 
for  three  pre-  repro- 
liminary  tests,  duction. 

Inductive  group 15.4  26.8 

M.  V.'s 2.4  1.9 

Deductive  Group 15.4  26.3 

M.  V.'s 2.4  1.8 

The  balance  of  advantage  seems  even  more  clearly 
on  the  side  of  the  group  taught  inductively.* 

(c)     Correlation  Between  Immediate  and  Deferred 
Reproduction. 

It  seems  likely  from  Table  IX,  already  given,  that 
there  is  considerable  positive  correlation  between 
the  results  of  immediate  reproduction  and  those  of 
deferred  reproduction.  That  is  to  say,  the  girls  who 
are  best  immediately  after  teaching  and  learning  are 
also  the  best  after  an  interval,  and  those  who  are 
worst  immediately  after  teaching  and  learning  re- 
main the  worst  after  some  time  has  elapsed.  But  in 
Table  IX  the  children  are  classified  on  the  basis  of 
their  marks  for  the  preliminary  tests,  and  this  classi- 
fication tends  to  obscure  much  of  the  correlation 
which  undoubtedly  exists.  In  the  following  tables 
the  classification  is  based  on  the  marks  obtained  in 
the  test  of  immediate  reproduction : 


*In  Tables  X  and  XI  the  difference  between  the  means  of  the 
work  of  the  two  groups  in  deferred  reproduction  is  about  twice  the 
'probable  error'  in  each  case,  even  on  the  assumption  that  the 
series  are  not  positively  correlated. 


66  INDUCTIVE   VS.   DEDUCTIVE    METHODS. 

Table  XII,  showing  the  results  for  Immediate  and  Deferred  Re- 
production compared,  of  the  Inductive  and  Deductive  Groups 
(positive  marks  only}. 

Deductive  Group. 

Marks  in  No. 

immediate                             of  Average  Marks  in  Reproduction, 

reproduction.                       girls.  First.  Second.  Third. 

Over  28 3  29.3                27.3  26.0 

28 5  28.0                25.8  24.8 

27,  26 5  26.2                 26.0  24.6 

Below  26 4  23.3                22.5  24.0 

Inductive  Group. 

Marks  in  No. 

immediate                            of  Average  Marks  in  Reproduction, 

reproduction.                       girls.  First.  Second.  Third. 

Over  28 5  29.4                28.8  27.0 

28 4  28.0                27.8  27.3 

27,  26 5  26.6                 25.0  25.2 

Below  26 3  23.0                23.7  22.7 

It  is  quite  obvious,  from  the  foregoing  table,  that 
considerable  positive  correlation  exists  between  im- 
mediate and  deferred  reproduction,  but  such  a  table 
gives  us  no  numerical  equivalent  for  correlation. 
The  correlation  coefficients  have,  however,  been 
worked  out,  and  for  the  Deductive  Group  the  coeffi- 
cient for  the  first  and  second  reproduction  is  +  .62 
(probable  error  .10),  and  for  the  second  and  third  is 
+  .58  (probable  error  .11). 

In  the  Inductive  Group  the  correlation  coefficient 
between  the  first  and  second  reproductions  is  +  .76 
(probable  error  .07),  and  between  the  second  and 
third  is  +  .76  (probable  error  .07).  All  the  figures 
indicate  high  reliability  for  the  results,  and  a  com- 
parison of  the  correlation  coefficients  for  the  Induct- 
ive and  Deductive  Groups  shows  the  work  of  the 
former  to  be  the  more  consistent. 


SECOND   SERIES   OF   EXPERIMENTS.  67 

(d)     Results  of  the  Test  on  New  Material. 

In  the  case  of  the  previous  school  we  found  that 
with  younger  children  of  a  lower  standard  the  de- 
ductive method  seemed  the  better  for  purely  repro- 
ductive purposes.  In  this  school  the  inductive  method 
seems  better,  even  for  purposes  of  reproduction. 
We  have  now  to  see  whether,  when  application  is 
made  to  new  material,  the  results  for  these  children 
agree  with  or  differ  from  those  of  the  preceding 
school.  First  let  me  show  the  results  for  the  two 
groups  as  wholes : 

Table  XIII,  showing  the  work  of  tJie  two  groups  compared  in  the 
Preliminary  Tests  and  in  the  Tests  of  Application  to  New 
Material. 

Average  Marks  for  New 
Material. 

When 
Average  mark  negative 

for  three  Positive  marks 

preliminary  marks  have  been 

tests.  only.  subtracted. 

Inductive  group 15.4  25.5  24.2 

M.  V.'s 2.4  2.7  3.4 

Deductive  group 15.4  23.3  21.9 

M.  V.'s 2.4  2.9  2.9 

We  have  a  clear  advantage,  in  both  cases,  on  the 
side  of  the  Inductive  Group.  The  difference  between 
the  averages  amounts  to  about  three  times  its  'prob- 
able error,'  even  on  the  assumption  that  the  series 
are  not  positively  correlated.  Once  again,  then,  we 
find  the  inductive  method  triumphant  when  applica- 
tion is  made  to  new  material.  Let  me  now  show  how 
far  this  is  a  difference  which  is  to  be  found  all  along 
the  line,  i.  e.,  for  the  weaker  as  well  as  for  the  abler 
pupils : 


68 


INDUCTIVE   VS.   DEDUCTIVE    METHODS. 


Table  XIV,  shoiving  the  work  of  the  two  groups  compared,  section 
1)11  section,  in  the  Preliminary  Tests  and  in  the  Test  of  Appli- 
cation to  New  Material  (positive  markJ,  and  positive  marks 
after  deduction  of  the  negative  marks). 

G  roup  A  ( Deductively        Group  B  ( Inductively 


Taught). 
Marks  for  New 

Taught). 
Marks  for  New 

Marks  for 

Material. 

Material. 

three 
preliminary 
tests. 
Over  50  

No. 
of 
girls. 
6 

(Posi- 
tive 
only). 
25.8 
23.7 
19.8 

(After 
de- 
duction). 
24.5 
21.7 
19.2 

No. 
of 
girls. 
0 
6 
5 

(Posi- 
tive 
only). 
27.3 
24.2 
25.0 

(After 
de- 
duction )  . 
26.0 
22.7 
24.0 

40  to  50  

6 

30  to  40.  . 

5 

There  seems  little  doubt  that  the  group  inductively 
taught  shows  a  superiority  which  is  general — a  supe- 
riority which,  somewhat  unexpectedly  to  me,  how- 
ever, seems  most  clearly  marked  in  the  weakest  (ini- 
tially considered)  of  the  three  sections  into  which 
each  group  is  divided. 


VL    THIRD  SERIES  OF  EXPERIMENTS. 
1.     General  Plan. 

As  in  the  previous  experiments,  a  whole  class  of 
children,  working  under  the  same  teacher,  with  the 
same  curriculum,  and  according  to  the  same  time- 
table of  work,  was  divided  into  two  equal  groups  on 
the  basis  of  several  tests  in  geometrical  definition, 
which  the  children  attempted  without  instruction  and 
without  help.  Then,  subsequently,  one  of  the  two 
groups  was  taught  inductively  and  the  other  group 
learnt  the  definitions.  There  were  tests  of  immedi- 
ate reproduction  immediately  after  the  lesson,  and 
""another  test,  which  might  also  be  called  a  test  of  im- 
mediate reproduction,  on  the  following  day.  About 
a  week  later  there  was  a  test  of  application  to  new 
material,  and,  three  weeks  after  this,  two  further 
tests  were  given,  which  will  be  referred  to  as  tests  of 
deferred  reproduction. 

The  work  was  done  with  fifty  children,  whose  av- 
erage age  was  9  years  3  months.  They  were  graded 
as  Standard  III  of  a  municipal  elementary  school  for 
boys,  situated  in  a  very  good  suburban  neighborhood 
in  the  southeast  of  London.  The  inductive  teaching 
was  done  in  this  case  not  by  me,  but  by  the  teacher  of 
the  class ;  whilst  the  group  which  studied  the  written 
definitions  was  taken,  during  that  particular  lesson, 
by  the  head  master  of  the  school.  All  the  tests  were 

69 


70  INDUCTIVE   VS.    DEDUCTIVE    METHODS. 

administered  by  the  class  teacher,  who  had  had  some 
experience  of  research  work  in  biology  as  well  as  in 
experimental  pedagogy.  One  of  the  boys'  fathers 
told  him  he  was  doing  Euclid  (which  he  wasn't),  and 
gave  him  a  'tip'  or  two  which  affected  some  of  his 
papers  adversely;  but,  with  that  exception,  the  suc- 
cess of  the  experiment  was  not  hindered  by  any  pre- 
vious knowledge  on  the  part  of  the  j  children. 
Whereas,  with  the  Standard  V  class  of  gijrls,  in  the 
experiment  just  described,  the  teacher'^  methods 
were  instructional  rather  than  either  definitely  in- 
ductive, deductive,  or  memoriter,  and  with  the  Stand- 
ards VI  and  VII  class  of  girls,  in  the  experiment 
which  has  just  been  described,  the  teacher's  methods^ 
were  both  inductive  and  memoriter,  according  to  the 
subject-matter  dealt  with;  in  this  third  case  the  re- 
action against  unintelligent  teaching  had  gone  so 
far  that,  whilst  the  inductive  teaching  was  extremely 
good,  the  memoriter  work  was  decidedly  novel  to  the 
children.  Novelty  has  a  stimulating  influence,  we  all 
know,  but  it  is  unlikely  that  its  influence  is  more  ef- 
fective in  result  than  that  of  habitual  practices.  In 
any  case  it  is  essential  to  try  the  experiment  with 
classes  differently  taught. 

2.     The  Preliminary  Tests  and  the  Method  of 
Marking. 

Just  as  before,  drawings  of  squares,  triangles,  ob- 
longs and  diameters  of  circles,  with  their  names  writ- 
ten against  the  drawings,  were  shown  to  the  children ; 
the  questions,  "  What  is  a  square?"  etc.,  were  written 
on  the  blackboard;  the  children  were  told  to  look  at 


THIRD   SERIES   OF   EXPERIMENTS.  71 

the  squares,  triangles,  etc.,  and  to  answer  the  ques- 
tions in  writing  as  well  as  they  could. 

The  units  of  marking,  as  before,  were  obtained 
after  a  careful  Review  of  the  answers  actually  given, 
and  it  was  found  that  the  units  previously  adopted 
were  quite  suitable.  A  few  instances  of  the  children  's 
attempts  at  spontaneous  definition  may  be  worth 
quoting.  It  must  be  remembered  that  these  children 
were  considerably  younger  than  either  of  the  classes 
of  girls  whose  work  has  previously  been  described, 
and  that  they  were  graded  as  Standard  III  as  com- 
pared with  Standards  V,  VI  and  VII.  On  the  other 
hand,  the  school  was  much  more  favorably  situated 
socially  than  either  of  the  schools  for  girls.  More- 
over, it  was  a  boys'  school;  and  boys,  whether 
through  greater  natural  ability  or  more  training,  are 
more  proficient,  geometrically,  than  girls. 

E.  D.,  aged  9  years  1  month,  wrote : 

1.  A  square  is  a  four  sided  figure  with  four  points  and  the  sides 
are  all  equal. 

2.  A  triangle  is  a  three  sided  figure  with  three  points  and  the 
sides  equal. 

3.  An  oblong  is  a  four  sided  figure  with  two  sides  long  and  two 
sides  short. 

4.  A  diameter  is  a  strait  line  that  goes  anything  like  a  circle 
and  will  go  across  any  way. 

If  we  mark  this  paper — E.  D.'s  first  preliminary 
test — on  the  system  of  marking  adopted  in  the  pre- 
vious experiments,*  we  see  that  for  his  definition  of 
a  square  he  receives  one  mark  for  ' figure,'  one  for 
the  adjective  ' sided,'  one  for  the  numerical  adjective 
'  four, '  and  one  for  the  equality  of  the  sides.  '  Points ' 
are  taken  as  equivalent  to  angles  or  corners,  and 


*The  reader  is  recommended  to  turn  to  page  28  for  the  list  of 
units. 


72  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

therefore  receives  a  mark,  whilst  the  numerical  ad- 
jective 'four'  also  scores.  This  gives  afotal  of  six 
marks  for  the  definition  of  the  square.  / 

The  definition  of  triangle  receives  pile  mark  for 
'  figure, '  one  mark  for  '  sided, '  one  matfk  for  '  three, ' 
one  for  *  points,'  and  another  for  'thrW'  (points). 
'The  sides  equal'  receives  a  mark  as  a  'bad  error,' 
but  there  were  so  few  of  these  in  the  preliminary 
tests  that  they  were  not  tabulated. 

The  boy's  definition  of  oblong  receives  a  mark  foi 
'figure,'  one  for  'sided,'  one  for  'four,'  one  for  'two 
sides  long,'  and  another  for  'two  sides  short.' 

His  last  definition  is  rather  weak.  He  obtains  a 
mark  for  'line'  and  one  for  'strait,'  and  that  is  all. 

When  one  remembers  that  these  are  untaught, 
spontaneous  definitions  given  by  a  boy  9  years  of 
age,  we  shall,  I  am  sure,  regard  them  as  affording 
evidence  of  considerable  ability.  Four  times  the  boys 
answered  these  questions  without  help  and  without 
criticism,  and  advanced  a  little  each  time.  This  is 
what  E.  D.  wrote  on  his  fourth  attempt — the  fourth 
preliminary  test — three  days  after  the  first : 

•      *•**»      i 

1.  A  square  is  a  four  skied  figure  with  four  equal  sides  and 
four  sharp  points. 

2.  A  triangle  is  a  three  sided  figure  with  three  equal  sides  to  it 
and  it  has  three  sharp  points, 

3.  An  oblong  is  a  four  sided  figure  with  four  points  but  the 
sides  are  not  all  the  same  two  sides  one  length  and  the  other  sides 
another  length. 

4.  The  diameter  of  a  circle  is  a  line  that  is  going  from  one  side 
to  the  other  side  of  the  circle  and  that  is  called  the  diameter  of  a 
circle  and  the  line  is  quite  strait. 

Let  us  see  how  far  this  fourth  paper  is  in  advance 
of  the  first.  The  definition  of  a  square  receives  the 
same  mark  as  before ;  it  is  slightly  more  concise  in 


THIRD    SERIES    OF    EXPERIMENTS.  73 

expression,  but  the  units  of  correct  description  are 
the  same  in  number  in  both  cases. 

The  definition  of  triangle  remains  unaltered. 

It  is  interesting  and  important  to  notice  that  even 
a  clever  boy  may  go  on  perpetrating  a  'bad  error ' 
unless  his  attention  is  drawn  to  it,  which,  of  course, 
the  conditions  of  the  experiment  did  not  permit  us 
to  do  in  these  tests. 

That  E.  D.  is  clever  for  a  nine-year-old  boy  is 
clearer  from  his  next  two  definitions  than  from  those 
of  the  square  and  triangle.  He  nearly  doubles  his 
previous  mark  for  his  definition  of  an  oblong.  He 
now  receives  marks  for  'points,'  for  'four'  (points), 
for  'two  sides  long'  and  for  'two  sides  short,'  and 
also  for  'two  long  sides  equal'  and  'two  short  sides 
equal. ' 

His  definition  of  diameter  has  also  improved.  He 
has  now  included  the  condition  that  it  must  go  from 
one  side  of  the  circle  to  the  other. 

These  papers  of  E.  D.  are  extremely  good  ones, 
and  do  not  represent  the  average  mark  of  the  class, 
which  ranges  from  11  to  13  units,  rather  than  from 
18  to  23,  which  E.  D.  obtains  for  his  first  and  fourth 
papers,  respectively. 

Let  me  now  give  examples  of  some  papers  below 
the  average. 

J.  C.,  aged  9  years  2  months,  answered  his  first  pre- 
liminary test  as  follows : 

1.  A  square  is  four  put  into  one  shape  with  equal  sides. 

2.  A  triangle  is  a  thing  that  has  no  equal  sides,  two  are  equal 
and  one  is  not,  and  it  has  three  sides. 

3.  An  oblong  is  not  a  square,  but  it  is  a  long  one 

4.  The  diameter  is  a  line  drawn  through  the  midle  of  a  circle. 

Side  by  side  with  this — J.  C.'s  first  preliminary 


74  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

test — let  us  compare  the  paper  worked  by  him  three 
days  later — his  fourth  preliminary  test : 

1.  A  square  is  a  shape  of  a  block  with  four  equal  sides. 

2.  A  triangle  is  a  long  square  with  only  three  sides,  the  two  side 
ones  are  both  the  same  and  the  top  one  is  not. 

3.  An  oblong  is  a  square  that  is  long,  with  two  equal  sides  and 
two  ends  which  are  not  the  same  size. 

4.  The  diameter  of  a  circle  is  a  line  drawn  down  the  midle. 

The  marks  for  the  definition  of  a  square  are  in 
both  cases  the  same :  ' shape'  receives  a  mark,  'sides' 
receives  one,  'four'  gets  one,  and  'equal'  (sides)  gets 
one. 

The  two  definitions  of  a  triangle  receive  the  same 
mark:  there  is  a  mark  for  'sides'  and  one  for  'three' 
(sides),  and  that  is  all. 

The  first  definition  of  an  oblong  receives  no  marks 
at  all,  whilst  the  one  given  later  receives  a  mark  for 
'sides,'  a  mark  for  'two  equal'  (sides),  and  one  for 
' '  two  ends  which  are  not  the  same  size  as  the  others. ' ' 

His  definition  of  a  diameter  remains  unchanged 
throughout  the  preliminary  tests ;  in  each  case  it  re- 
ceives two  marks  only,  one  for  'line'  and  one  for 
'  drawn  through  the  middle. ' 

J.  C.  advances  from  a  mark  of  8  in  the  first  test  to 
11  in  the  fourth.* 

Having  given  some  indications  of  the  work  done  in 
the  Preliminary  Tests,  on  the  results  of  which  the 
class  was  divided  into  two  equal  groups,  let  me  set 
out  in  detail  the  chronological  progress  of  the  experi- 
ment. 


*The  average  improvement  from  test  to  test  is  shown  on  page  91. 


THIKD   SEEIES   OF   EXPEKIMENTS.  75 

3.     Chronology  of  the  Experiment. 

A  first  Preliminary  Test  was  given  at  9.40  A.  M. 
on  Monday,  October  23,  1911,  immediately  after 
Scripture  lesson;  a  second  on  Tuesday,  October  24, 
a  third  on  Wednesday,  October  25,  and  a  fourth  on 
Thursday,  October  26,  at  the  same  hour  and  after  the 
same  lesson  on  each  occasion.  On  the  results  of  these 
four  tests  the  class  was  divided  into  two  equal 
groups. 

On  Thursday,  November  9,  at  9.40  A.  M.,  one  of 
the  two  groups  was  taught  the  definitions  inductively 
by  the  methods  already  described,  whilst  the  other 
studied  them  as  written  out,  with  reference  to  the 
drawings  appended  to  the  verbal  descriptions. 
Twenty-three  minutes  were  taken  by  the  teacher  to 
teach  the  definitions  inductively;  the  same  time  was 
allowed  to  the  group  which  was  studying  the  defini- 
tions with  a  view  to  remembering  them.  Both 
groups  of  children  were  aware  that  they  were  to  be 
tested  on  their  work  at  the  close  of  the  lesson.  Ac- 
cordingly, at  10.15  A.  M.,  a  test  was  given  in  immedi- 
ate reproduction.  In  this  school,  since  the  children 
were  young  and  the  exercises  very  novel,  we  thought 
it  best  to  take  another  test,  identical  with  the  test  of 
immediate  reproduction,  at  the  same  hour  on  the 
next  day,  Friday,  November  10,  to  see  how  far  the 
first  day's  test  was  reliable.  These  two  tests  will  be 
referred  to  as  the  two  Tests  of  Immediate  Repro- 
duction. 

At  9.40  A.  M.  on  Thursday,  November  16,  a  test 
was  given  on  the  application  of  what  had  been  learnt 
to  new  analogous  material  with  the  object  of  discov- 


76  INDUCTIVE   VS.   DEDUCTIVE    METHODS. 

ering  which  of  the  two  groups  attacked  the  new  mate- 
rial the  more  successfully. 

Finally,  two  tests  of  deferred  reproduction  were 
given  at  9.40  A.  M.  on  Thursday,  December  7,  and 
Friday,  December  8.  The  children  were  quite  una- 
ware that  they  would  be  required  to  take  any  of  these 
tests,  with  the  exception  of  the  one  immediately  after 
the  teaching  and  learning  on  Thursday,  November  9. 

4.     The  Final  Tests  and  the  Method  of  Marking. 

The  two  tests  of  Immediate  Reproduction  were 
repetitions  of  the  Preliminary  Tests,  as  were  also 
the  two  tests  of  Deferred  Eeproduction.  The  units 
of  marking  previously  used  in  the  Preliminary  Tests 
were  found  quite  satisfactory.  The  tests  of  deferred 
reproduction  received  negative  as  well  as  positive 
marks.  One  or  two  specimens  of  the  worked  papers 
may  be  of  interest. 

L.  0.,  aged  9  years,  who  scored  13, 16, 18,  18  in  his 
preliminary  tests,  and  was  taught  inductively,  for  his 
first  test  of  Immediate  Eeproduction  on  November 
9,  wrote  as  follows : 

1.  A  square  is  a  shape  with  four  lines  all  the  same  size  and  for 
corners  all  the  same  size. 

2.  A  triangle  is  a  shape  with  three  corners  and  three  lines. 

3.  A  oblong  is  a  shape  with  two  long  lines  the  same  size,  and 
two  shorter  lines  the  same  size. 

4.  A  diameter  of  a  circle  is  a  line  which  goes  from  one  part  to 
the  opposite  part  touching  the  middle  of  the  circle  and  keeps  inside 
the  circle. 

This  is  a  good  set  of  answers  for  a  boy  only  nine 
years  of  age.  Marked  on  the  system  of  units  previ- 
ously used,  the  definition  of  a  square  receives  seven 
marks,  the  definition  of  a  triangle  receives  five 


THIKD    SERIES    OF    EXPERIMENTS.  77 

marks,  that  of  the  oblong  receives  seven  marks,  and 
that  of  the  diameter  of  a  circle  receives  three  marks. 
It  will  be  seen  that,  compared  with  the  standard  defi- 
nitions, there  is  a  loss  of  one  mark  in  the  definition 
of  a  square,  since  the  description  '  straight'  is  not 
applied  to  the  ' lines'  or  'sides.'  The  definition  of 
triangle  is  correct. 

Six  marks  are  lost  on  the  oblong.  *  Four  equal  cor- 
ners'  are  omitted,  carrying  three  marks.  ' Straight' 
is  omitted  in  describing  the  lines  or  sides,  and  the  two 
long  lines  and  the  two  short  lines  are  not  described 
as  opposite. 

One  mark  only  is  lost  on  the  definition  of  '  diam- 
eter;' the  line  is  not  described  as  ' straight.'  The 
marks,  totaled,  amount  to  22. 

On  the  next  day's  test  L.  0.  goes  down  one  mark. 
His  definitions  of  square  and  triangle  remain  un- 
changed. In  his  definition  of  oblong  he  omits  the  two 
points  previously  inserted,  namely,  that  the  two  long 
lines  are  of  the  same  length,  and  that  the  two  short 
lines  are  of  the  same  length.  But  in  the  definition 
of  the  diameter  of  a  circle  he  inserts  the  description 
'  straight '  which  he  had  omitted  the  day  before  alto- 
gether. He  thus  scores  21  marks  for  his  second  test. 

Let  us  now  see  what  happens  a  month  later  when 
the  same  test  is  applied  a  third  time.  I  give  his 
paper  in  full. 

L.  0.,  aged  9  years  1  month,  on  December  7,  1911, 
in  his  first  test  of  Deferred  Eeproduction,  wrote  as 
follows : 

1.  A  square  is  a  shape  with  four  lines  all  the  same  length,  arid 
four  corners  all  the  same  size. 

2.  A  triangle  is  a  shape  with  three  lines  and  three  corners. 

3.  An  oblong  is  a  shape  with  four  lines  two  long  lines  both  the 
same  size,  and  two  shorter  lines  both  the  same  length. 


78  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

4.  A  diameter  of  a  circle  is  a  line  inside  which  goes  from  one 
part  of  the  circle  and  touches  the  middle  of  the  circle  goes  on  to 
the  opposite  part  of  the  circle  to  where  it  started  and  it  must  be  a 
straight  line. 

This  is  a  very  good  paper,  and  scores  a  total  of  23 
marks,  an  advance  on  the  work  of  the  month  before. 
On  the  day  following,  on  which  was  given  the  second 
test  of  Deferred  Eeproduction,  L.  0.  scored  24 
marks,  for  the  description  ' straight'  of  the  sides  of 
the  square,  omitted  on  December  7,  was  included  on 
December  8.  His  average  mark  for  his  two  tests  of 
Immediate  Eeproduction,  those,  namely,  of  the  9th 
and  10th  of  November,  was  21.5 ;  his  average  mark 
for  his  two  tests  of  Deferred  Eeproduction  was  23.5. 

It  must  not  be  thought  that  every  boy  obtains  more 
marks  a  month  after  the  lesson  than  he  does  for  his 
immediate  tests,  but  many  of  them  do ;  and  the  aver- 
age result  shows  only  a  slight  decline,  rather  more 
marked  in  the  group  taught  deductively  than  in  the 
group  taught  inductively.  This  is  explained  by  the 
fact  that  both  the  teaching  and  the  learning  were  well 
within  the  comprehension  of  the  boys.  When  this  is 
the  case,  and  they  work  in  consequence  with  interest 
and  enthusiasm,  they  forget  surprisingly  little. 

It  may  now  be  of  some  value  if  I  give  the  corre- 
sponding papers  of  a  boy  in  the  Deductive  Group. 

E.  S.,  aged  9  years  2  months,  who  scored  14,  19, 19, 
18  marks  in  his  four  preliminary  tests,  in  his  first 
test  of  Immediate  Eeproduction  wrote : 

1.  A  square  is  a  shape  with  four  sides  and  four  corners.    The 
sides  are  straight  and  all  the  same  length.    The  corners  are  all  the 
same  size. 

2.  A  triangle  is  a  shape  with  three  sides  and  three  corners. 

3.  An  oblong  is  a  shape  with  four  sides  and  four  corners.    The 
sides  are  straight  and  there  are  two  long  sides  and  two  short  sides. 


THIRD    SERIES    OF    EXPERIMENTS.  79 

The  long  sides  are  opposite  one  another  and  are  the  same  length, 
and  the  two  short  sides  are  opposite  and  are  the  same  length. 

The  diameter  of  a  circle  is  a  straight  line  which  goes  through 
the  centre  of  the  circle. 

Only  two  units  of  definition  are  omitted :  the  equal- 
ity of  the  angles  is  left  out  in  the  definition  of  the 
oblong,  and  the  delimitation  of  the  diameter  by  the 
opposite  parts  of  the  circumference  of  the  circle  is 
omitted  in  the  last  definition.  It  is  an  excellent  pa- 
per, appearing  on  the  face  of  it,  if  one  compares  it 
with  the  verbal  expression  of  the  definitions  which 
were  given  to  be  studied,  to  owe  a  great  deal  to  a 
highly  developed  rote  memory.  If  that  is  so,  it  is 
memory  for  statements  that  are  really  understood, 
since  they  persist  unchanged  without  the  subsequent 
intrusion  of  stupid  errors,  and  an  unusually  high 
mark  is  obtained  by  this  boy  for  his  power  of  appli- 
cation to  new  material.  I  propose  to  defer  consid- 
eration of  the  latter  issue,  since  just  now  we  are  con- 
cerned only  with  the  tests  of  Immediate  and  De- 
ferred Reproduction. 

The  next  day  E.  S.  obtained  29  marks,  as  compared 
with  28  of  the  previous  day.  There  were  slight 
changes  of  verbal  expression.  For  instance,  the  tri- 
angle became  "a  three  cornered  figure  with  three 
sides."  The  equality  of  the  angles  was  omitted  in 
the  square,  but  on  this  occasion,  though  not  in  the 
previous  test,  was  included  in  the  definition  of  the 
oblong.  In  the  definition  of  the  circle  an  improve- 
ment was  shown;  the  point  was  included  which 
the  day  before  had  been  omitted;  it  was  now  men- 
tioned that  the  line  went  from  ' '  one  side  to  the  oppo- 
site side  of  the  circle." 

One  month  later  E.   S.   scored  28  marks.     He 


80  INDUCTIVE   VS.    DEDUCTIVE    METHODS. 

omitted  the  description  ' straight'  in  his  definition  of 
a  diameter  of  a  circle  which  he  had  before  included. 
On  the  day  following  he  made  the  same  omission. 
Otherwise  his  definitions  are  just  as  good  as  those 
which  he  had  given  a  month  before.  His  average 
mark  for  Immediate  Reproduction  is  28.5,  and  for 
Deferred  Reproduction  is  28.0. 

Let  me  give  one  more  illustration,  the  work  of  a 
boy  who  obtained  6,  6,  7  and  7  marks  in  his  four  pre- 
liminary tests,  and  who  also  was  taught  in  the  De- 
ductive Group. 

H.  W.,  aged  10  years  1  month,  in  his  first  test  of 
Immediate  Reproduction  wrote : 

1.  A  square  is  a  shape  with  four  corners  and  four  sides  the 
same  size. 

2.  A  triangle  is  a  shape  with  three  corners  and  three  sides. 

3.  An  oblong  is  a  shape  with  two  small  sides,  and  two  big  sides 
opposite  one  another. 

4.  The  diameter  of  a  circle  is  a  line  passing  through  the  middle 
of  the  circle. 

Marked  on  the  same  units  as  before,  the  definition 
of  a  square  obtains  six  marks ;  the  definition  of  tri- 
angle obtains  five  marks ;  the  definition  of  oblong  re- 
ceives six  marks,  for  it  is  called  a  ' shape,7  its  'four' 
'sides'  are  implied,  its  'two  long  sides'  and  its  'two 
short  sides'  are  noted,  and  the  fact  that  its  'two  long 
sides  are  opposite  each  other.'  The  definition  of 
diameter  receives  two  marks.  This  is  not  a  strong 
paper ;  it  scores  19  marks  only  as  a  total,  but  it  im- 
plies a  very  great  advance  on  this  boy's  preliminary 
tests.  One  point  of  interest  lies  in  this.  Whereas, 
in  the  papers  of  R.  S.,  recently  given,  there  was  an 
appearance  of  rote  learning  in  the  answers,  there  is, 
in  the  case  of  this  boy,  no  direct  indication  of  that. 

H.  W.'s  next  test  of  Immediate   Reproduction, 


THIKD   SERIES   OF    EXPERIMENTS.  81 

worked  on  the  following  day,  receives  the  same  num- 
ber of  positive  marks,  namely,  19.  A  'bad  error'  has 
crept  in,  for  the  corners  of  the  triangle  are  described 
as  all  the  same  size.  The  'two  small  sides'  of  the 
oblong  are  now  called  "two  small  tops,"  but  this  and 
the  'bad  error'  are  the  only  changes.  One  month 
later,  for  his  first  test  of  Deferred  Eeproduction, 
H.  W.  wrote : 

1.  A  square  is  a  shape  with  four  corners  and  four  sides  opposite 
one  another  and  they  are  all  of  the  same  length. 

2.  A  triangle  is  a  shape  with  three  corners  and  three  sides  they 
are  not  opposite  one  another. 

3.  An  oblong  is  a  shape  with  three  corners  and  three  sides,  it  is 
a  zig-zag  shape  not  all  the  same  length. 

4.  A  diameter  of  a  circle  is  a  line  passing  through  the  middle 
of  it 

Considerable  changes  are  evident  in  this  paper. 
There  are,  as  before,  six  positive  units  of  correct 
description  in  the  definition  of  the  square;  but  the 
statement  "four  sides  opposite  one  another"  has 
been  adjudged  a  'bad  error.'  It  is,  of  course,  the 
confused  application  of  some  phrase  remembered, 
but  not  understood.  Let  it  not  be  supposed,  however, 
that  no  child  inductively  taught  makes  similar  errors. 

The  definition  of  triangle  receives  five  marks  as 
before.  The  memory  of  the  oblong  has  largely  gone. 
It  is  still  remembered  that  it  is  a  'shape'  and  has 
'corners'  and  'sides,'  and  thus  three  positive  marks 
are  obtained.  But  to  give  an  oblong  'three'  corners 
and  'three'  sides  and  to  call  it  'zig-zag'  shape  is  held 
to  involve  three  bad  errors.  The  definition  of  diam- 
eter remains  unchanged,  and  scores  two  marks.  The 
paper  as  a  whole  receives  16  positive  marks,  and 
there  are  four  marks  for  bad  errors. 


82  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

But  on  the  next  day,  in  his  second  test  of  Deferred 
Eeproduction,  H.  W.  made  a  decided  recovery.  He 
then  wrote : 

1.  A  square  is  a  shape  with  four  corners  and  four  sides  they 
are  opposite  one  another,  with  all  the  sides  and  corners  an  equal 
size. 

2.  A  triangle  is  a  shape  with  three  sides  and  corners  it  is  a 
zig-zag  shape. 

3.  An  oblong  is  a  shape  with  four  sides,  two  long  sides  and  two 
short  tops  they  are  opposite  one  another. 

4.  A  diameter  of  a  circle  is  a  line  passing  through  the  middle 
of  it. 

This  is  undoubtedly  H.  W.  's  best  paper.  He  scores 
the  highest  marks  he  has  yet  scored  for  the  definition 
of  the  square,  namely,  seven  positive  marks,  since, 
for  the  first  time,  he  has  mentioned  the  equality  of 
the  corners,  but  he  retains  his  '  bad  errors. '  The  defi- 
nition of  a  triangle  remains  unchanged  in  correct 
units ;  it  is  held  inadmissible  to  call  the  triangle  a  zig- 
zag shape.  The  definition  of  oblong  has  returned  to 
its  first  condition;  indeed,  it  is  rather  better,  for  it 
is  easier  now  to  regard  H.  W.  as  implying  that  the 
6 two  long'  and  'two  short'  sides  are  pairs  of  equals. 
The  mark  for  the  double  equality  is,  however,  not 
given,  as  the  meaning  is  somewhat  doubtful.  The 
definition  of  diameter  remains  unchanged.  H.  W. 
scores  20  positive  marks  for  his  paper  and  one  nega- 
tive mark  for  a  'bad  error. '  Again  we  find  the  marks 
for  Deferred  Eeproduction  not  much  inferior  to 
those  of  Immediate  Eeproduction  in  this  case;  in- 
deed, the  last  paper  is  the  best  the  boy  did  through- 
out the  series. 

I  trust  that  the  inclusion  of  these  papers  will  be  of 
service  in  giving  life  and  body  to  the  rather  bloodless 
array  of  figures,  which  I  give  subsequently,  dealing 


THIRD   SERIES   OF   EXPERIMENTS.  83 

with  the  results  of  the  tests  in  Immediate  and  De- 
ferred Eeproduction. 

The  Test  of  Application  to  New  Material  was  iden- 
tical with  that  used  in  the  experiment  previously  de- 
scribed. Drawings  of  rhombuses,  trapeziums,  rhom- 
boids and  diagonals  of  squares,  with  their  names 
appended,  were  shown  to  the  children,  and  they  were 
required  to  answer  in  writing  the  questions:  "What 
is  a  rhombus  ? ' '  etc.  It  may  add  to  the  facility  with 
which  the  progress  of  this  experiment  is  understood 
if  I  give  verbatim  one  or  two  of  the  worked  papers. 
In  the  test  of  application  to  new  material  negative 
marks  were  assigned  as  well  as  positive  marks. 

L.  0.,  aged  9  years,  a  boy  who  worked  in  the  In- 
ductive Group,  whose  work  in  Immediate  and  De- 
ferred Eeproduction  has  already  been  quoted,  wrote 
the  following  paper  in  this  test : 

1.  A  rhombus  is  a  shape  containing  four  lines  all  the  same 
length,  so  that  if  you  looked  at  it  one  way  it  seems  to  bend  back- 
ward, and  if  you  look  at  it  again  it  looks  to  bend  forward. 

2.  A  rhomboid  is  a  shape  also  containing  four  lines,  two  long 
lines   both  the  same  length,  and  two  shorter  lines  both  the  same 
length. 

9.  A  trapezium  is  a  shape  with  four  lines  three  long  ones,  and 
one  short  one. 

4.  A  diagonal  of  a  square  is  a  str eight  line  going  from  one  cor- 
ner to  its  opposite  one. 

L.  0.  receives  four  marks  for  his  definition  of 
rhombus — one  for  *  shape,'  one  for  'lines,'  one  for 
'  four, '  and  one  for  all  the  same  '  length. '  He  receives 
three  marks  for  his  definition  of  trapezium,  one  for 
t  shape,'  one  for  *  lines,'  and  one  for  'four.'  His 
statement  that  there  are  three  long  lines  and  one 
short  one  was  not  held  to  be  equivalent  to  the  state- 
ment that  the  sides  were  unequal,  but  it  was  not  con- 
sidered a '  bad  error. '  For  his  definition  of  rhomboid 


84  INDUCTIVE   VS.    DEDUCTIVE    METHODS. 

he  obtains  seven  marks — one  for  '  shape/  one  for 
'lines/  one  for  'four,'  one  for  'two  long  lines,'  one 
for  '  two  shorter  lines, '  and  two  for  the  pair  of  equal- 
ities in  the  length  of  the  lines. 

The  definition  of  diagonal  receives  four  marks,  one 
for  '  line, '  one  for  * '  streight, ' '  and  two  for  ' '  from  one 
corner  to  its  opposite  one." 

The  paper  scores  a  total  of  18  positive  marks,  and 
there  are  no  'bad  errors ;'  the  average  mark  obtained 
by  the  boys  of  the  Inductive  Group  is  rather  lower 
than  this. 

E.  S.,  aged  8  years  5  months,  who  also  was  taught 
in  the  Inductive  Group,  wrote : 

1.  A  rhombus  is  a  figure  with  four  straight  sides  and  four 
equal  corners. 

2.  A  trapezium  is  a  figure  with  four  corners  which  are  equal 
with  four  sides. 

3.  A  rhomboid  is  a  figure  with  two  small  sides  which  are  hori- 
zontal and  two  bigger  parlerlell  lines  equal. 

4.  A  diagonal  of  a  square  is  a  straight  line  from  one  corner  to 
another  corner. 

The  first  definition  receives  a  mark  for  'figure,'  a 
mark  for  'sides,'  one  for  'four,'  one  for  'straight,' 
one  for  'corners,'  and  one  for  'four;'  a  total  of  six 
positive  marks;  but  there  is  one  'bad  error' — the 
corners  are  not  equal :  boys  taught  inductively  can 
obviously  make  the  same  sort  of  blatant  error  as 
boys  taught  deductively. 

The  second  definition  receives  a  mark  for  '  figure, ' 
one  for  '  corners, '  one  for  '  four, '  one  for  '  sides, '  and 
one  for  'four'  (sides) ;  a  total  of  five  positive  marks ; 
but,  again,  there  is  a  'bad  error' — the  angles  of  the 
trapezium  are  not  equal.  The  definition  of  rhomboid 
obtains  seven  positive  marks — one  for  'figure,'  one 
for  'sides,'  one  for  'two  small'  (sides),  one  for  'two 


THIRD   SERIES   OF    EXPERIMENTS.  85 

bigger'  lines,  a  mark  for  saying  the  two  bigger  are 
equal,  and  one  for  saying  the  two  bigger  lines  are 
parallel.  There  is  one  'bad  error;'  the  two  small 
sides  were  in  one  case  only  drawn  horizontally.  The 
definition  of  diagonal  receives  three  positive  marks 
— one  for  *  line,'  one  for  '  straight,'  and  one  for  "  from 
one  corner  to  another : ' '  the  further  specification  of 
' opposite'  corner  is  omitted. 

The  paper,  as  a  whole,  gains  21  positive  marks, 
with  three  negative  marks  for  'bad  errors.' 

Let  me  quote  one  more  illustration  from  among 
the  boys  who  were  taught  inductively. 

H.  B.,  aged  9  years  2  months,  wrote : 

1.  A  rhombus  is  a  fugare  which  is  like  a  square  and  has  fore 
corners. 

2.  A  trapezium  is  something  like  a  triangle  only  it  has  fore 
corners. 

3.  A  rhomboid  is  a  sought  of  fugare  which  is  something  like 
an  oblong. 

4.  A  diagonal  of  a  square  is  a  square  with  a  line  across  the 
midal. 

This  is  a  very  weak  paper ;  it  was  worked  by  a  boy 
who  was  almost  at  the  bottom  of  the  Inductive 
Group  in  the  preliminary  tests,  and  he  seemed  to  jus- 
tify his  position.  It  is  psychologically  interesting 
that  he  apprehended  the  similarity  between  the  work 
now  required  and  the  work  he  had  been  taught,  but 
was  unable  to  specify  the  points  of  similarity  and 
difference  between  the  figures  of  the  first  set  and  the 
figures  of  the  second  set.  He  had  but  little  know- 
ledge and  could  not  apply  much  of  that.  His  marks 
are :  three  for  his  definition  of  rhombus,  two  for  his 
definition  of  trapezium,  one  for  his  definition  of 
rhomboid,  and  one  for  the  definition  of  the  diagonal 
of  a  square.  "Across  the  midal"  is  not  held  to  be 


86  INDUCTIVE   VS.   DEDUCTIVE    METHODS. 

wrong,  though  it  might  be ;  in  any  case  it  is  not  re- 
garded as  sufficiently  definite  to  obtain  a  mark.  It 
is  regarded  as  a  'bad  error'  to  say  that  the  diagonal 
of  a  square  is  a  square.  H.  B.  receives  a  total  of 
seven  positive  marks,  with  one  negative  mark  for 
bad  errors. 

It  is,  perhaps,  worthy  of  note  that  this  boy  falls 
from  17  in  his  test  of  Immediate  Eeproduction  to 
9  in  his  test  of  Deferred  Reproduction.  He  can- 
not apply  his  old  knowledge  and  he  cannot  remember 
it  for  more  than  a  day  or  two. 

Let  us  now  turn  to  some  illustrative  examples  of 
the  work  of  the  group  taught  deductively. 

G.  M.,  aged  8  years  1  month,  wrote : 

1.  A  rhombus  is  a  figure  with  two  slanting  sides   and  two 
straight  ones  arranged  so  that  two  of  the  sides  are  facing  each 
other  and  the  other  two  opposite  each  other  and  also  four  corners. 

2.  A  trapezium  is  a  figure  with  four  slanting  sides  arranged  so 
that  there  are  two  sides  nearly  the  same  length,  these  two  are 
generally  touching  each  other.     Then  there  is  a  smaller  one  and 
yet  a  smaller  one  still,  so  that  there  are  four  sides  and  two  equal 
ones  the  others  ofcourse  are  not. 

3.  A  rhomboid  is  a   figure  with  two  slanting  sides  and  two 
straight  ones  and  also  four  corners  two  of  the  sides  are  longer 
than  the  other  two  and  also  are  opposite  one  another  and  so  are  the 
two  shorter  sides.    There  can  be  ones  upright  and  lying<  down  and 
also  slanting  ones. 

4.  A  diagonal  of  a  square  is  a  line  drawn  from  one  corner  to 
the  other  it  need  not  have  to  be  drawn  from  a  corner  for  it  could 
be  from  the  middle  of  the  top  to  the  middle  of  the  bottom,  but  you 
can't  have  it  so  that  it  is  from  the  middle  of  the  one  side  to  the 
middle  of  the  bottom  or  to  the  middle  of  the  top.    For  the  diameter 
is  the  greatest  and  longest  line  you  can  have  across  it  or  down  it 
and  that  wouldn't  be  the  longest,  not  nearly. 

This  is  an  excellent  paper  for  a  boy  of  eight  years 
of  age.  He  was  taught  in  the  Deductive  Group,  but 
evidently  he  is  quite  capable  of  applying  what  he  has 
learnt.  It  would  be  a  serious  error  to  suppose  that 
because  a  boy  has  learnt  a  set  of  definitions  therefore 


THIRD   SEKIES   OF    EXPERIMENTS.  87 

he  cannot  apply  them.  In  a  very  large  number  of 
cases  he  certainly  can.  The  contention  raised  in  this 
monograph  is  that  inductive  teaching  produces  a 
higher  transfer  to  new  material  than  deductive,  not 
that  deductive  teaching  involves  no  transfer  at  all. 
This  first-rate  paper  may  do  something  to  prevent 
an  exaggerated  conclusion  which  the  subsequent  fig- 
ures may  not  succeed  in  adequately  moderating.  Let 
us  mark  the  paper  on  the  usual  system  of  units.  G. 
M.  is  evidently  using  the  word  straight  to  mean,  as  it 
often  does  with  boys,  horizontal  and  vertical ;  he  does 
not  mean  that  only  two  of  the  lines  are  ' straight'  in 
the  proper  sense.  And  he  is  wrong  on  his  own  mean- 
ing, for  one  of  the  rhombuses  drawn  had  neither  ver- 
tical nor  horizontal  lines,  but  two  of  them  had,  and 
to  these  he  has  apparently  confined  his  attention. 
He  receives  a  mark  for  'figure,'  a  mark  for  *  sides,' 
and  one  for  'four'  (sides),  which  is  involved  in  his 
pair  of  twos,  and  one  for  'corners'  and  one  for  'four.' 
He  gets  two  marks  for  seeing  that  the  opposite  sides 
are  paired.  This  marking  yields  a  total  of  seven  pos- 
itive marks,  whilst  he  receives  a  negative  mark  for 
being  wrong  on  his  own  meaning  of  'straight.'  In 
his  definition  of  trapezium  he  receives  a  mark  for 
'figure,'  one  for  'sides,'  and  one  for  'four.'  His  first 
description  of  the  sides  is  held  to  be  equivalent  to 
saying  they  are  unequal,  so  he  receives  a  mark  for 
that.  Later  he  is  marked  for  a  'bad  error'  in  saying 
that  two  of  the  sides  are  equal.  They  are  so  in  one 
of  the  trapeziums  only.  For  the  definition  of  trape- 
zium, then,  he  gets  four  positive  marks,  with  one  neg- 
ative mark  for  a  'bad  error.'  Again,  in  his  definition 
of  a  rhomboid  we  find  a  misuse  of  the  word  straight, 
and  again  he  is  wrong,  even  on  his  own  meaning. 


88  INDUCTIVE   VS.    DEDUCTIVE    METHODS. 

But  he  obtains  positive  marks  for  '  figure, '  for  *  sides, ' 
for  'four'  (sides),  for  'two  long'  (sides),  for  'two 
shorter'  (sides),  and  two  marks  for  noting  the  pairs 
of  opposites.  He  also  notes  the  'four  corners.'  He 
thus  receives  nine  positive  marks  and  one  mark  for  a 
'bad  error.' 

His  definition  of  diagonal  is  extremely  interesting. 
He  receives  two  positive  marks  only — one  for  'line' 
and  one  for  "from  one  corner  to  another."  After 
that,  alas!  the  transfer  from  diameter  (the  corre- 
sponding definition  which  was  learnt)  has  been  too 
thorough.  No  diameters  were  drawn  in  the  squares 
which  were  before  the  boy's  eyes,  and  it  is  not  unfair 
to  call  the  lapse  into  diameter  a  'bad  error.'  This 
definition  receives  therefore  two  positive  marks  and 
one  negative  mark.  The  paper,  as  a  whole,  receives 
a  total  of  22  positive  marks,  and  there  are  four  bad 
errors;  it  is  considerably  above  the  average  of  the 
papers  worked  by  the  Deductive  Group  generally. 

H.  W.,  aged  10  years  1  month,  whose  work  in  Im- 
mediate and  Deferred  Eeproduction  has  already 
been  quoted,  wrote  the  following  in  his  test  of  appli- 
cation to  new  material  : 

1.  A  rhombus  is  a  shape  something  like  the  shape  of  a  diamond. 

2.  A  trapezium  is  a  shape  with  four  corners  not  opposite  one 
another  their  are  different  shapes  of  trapeziums  they  are  a  zig- 
zag shape  some  corners  longer  than  others,  they  are  not  squares. 

3.  A  rhomboid  is  a  shape  with  two  small  tops  both  opposite  one 
another,  and  with  two  long  sides  with  the  corners  exactly  opposite 
one  another. 

4.  A  diagonal  of  a  square  there  is  a  square  and  a  line  passes 
right  through.    Sometimes  they  pass  from  side  to  side  other  times 
from  corner  to  corner. 

H.  W.'s  definition  of  rhombus  receives  one  mark 
only — a  mark  for  the  description  'shape.'  For  the 
definition  of  trapezium  three  positive  marks  are 


THIKD   SERIES   OF    EXPERIMENTS.  89 

gained — one  for  *  shape, '  one  for  '  corners, '  and  one 
for  'four.'  There  are  no  'bad  errors.'  It  was  not 
thought  admissible  to  regard  the  expression  "some 
corners  longer  than  others"  as  involving  the  ine- 
quality of  the  angles.  His  definition  of  a  rhomboid 
receives  a  mark  for  'shape,'  one  for  'sides,'  one  for 
'corners,'  and  one  for  'four'  sides,  for  the  number  of 
sides  is  involved  in  the  rest  of  his  answer.  He  also 
receives  a  mark  for  "two  small  tops,"  one  for  "two 
long  sides,"  and  one  for  noting  that  the  two  small 
sides  are  'opposite'  each  other.  The  opposition  of 
the  angles  has  not  been  allowed  for  in  the  system  of 
marking.  This  definition  therefore  receives  a  total 
of  seven  positive  marks.  The  definition  of  diagonal 
receives  two  positive  marks  only — one  for  'line'  and 
one  for  'from  corner  to  corner.'  It  was  regarded  as 
a  bad  error  to  say  that  "sometimes  they  pass  from 
side  to  side. ' '  The  total  marks  for  this  paper  amount 
to  13  positive  marks,  from  which  one  has  to  be  de- 
ducted for  'bad  errors.' 

Let  me  now  pass  to  the  work  of  a  boy  who  scored 
13,  12,  12  and  11  in  his  four  preliminary  tests.  It 
seems  likely  from  these  figures  that  we  are  dealing 
with  a  boy  of  little  educability,  and  this  suggestion  is 
confirmed  by  his  later  work.  In  his  two  tests  of  Im- 
mediate Reproduction  he  scores  an  average  of  18.5 
marks;  in  both  tests  of  Deferred  Reproduction  he 
scores  13  marks,  so  that  a  month  afterwards  he  is 
back  again  to  the  position  he  occupied  before  he 
learnt  the  definitions,  and  he  completely  fails  in  ap- 
plying what  he  has  learnt. 

A.  R.,  aged  8  years  6  months,  the  boy  whose  work 
has  just  been  described  generally,  wrote : 

1.     A  rhombus  is  a  square  which  is  not  strate  up. 


90  INDUCTIVE   VS.   DEDUCTIVE    METHODS. 

2.  A  trapezium  is  a  four-sided  thing  which  sides  are  not  all 
strate. 

3.  A  rhomboid  is  like  an  oblong  but  its  lines  are  not  strate  up. 

4.  A  diagonal  of  a  square  is  a  diameter  of  a  circle  only  it  is 
a  squear. 

A.  B.  has  seen  some  general  resemblance  between 
the  *  figures'  of  his  first  set  of  definitions  and  those  of 
his  second  set,  but  the  resemblances  have  hindered 
rather  than  helped  him,  for  a  rhombus  is  not  a 
square,  and  a  diagonal  of  a  square  is  not  a  diameter 
of  a  circle.  The  meaning  of  the  word  "strate"  is 
misconceived ;  his  reference  to  the  sides  of  the  trape- 
zium is  not  incorrect  on  the  basis  of  his  own  mean- 
ing. Of  positive  marks,  on  the  system  of  marking 
adopted,  he  can  obviously  obtain  very  few.  He 
scores  no  marks  for  his  definition  of  rhombus,  two 
for  his  definition  of  trapezium,  one  for  his  definition 
of  rhomboid,  and  none  for  his  definition  of  diagonal. 
His  three  positive  marks  are  subject  to  a  deduction 
of  two  for  the  'bad  errors'  previously  indicated. 
Boys  of  this  kind  are  the  despair  of  the  teacher,  but 
the  evidences  yielded  by  his  work  do  not  point  so 
much  to  stupidity  as  to  ineducability. 

Possibly  the  reader  may  already  have  gathered 
from  a  perusal  of  the  papers  which  I  have  used  as 
illustrations  some  opinions  of  his  own  as  to  the  rela- 
tive applicability  of  the  two  methods  of  teaching  and 
learning.  But  all  such  opinions  need  to  be  confirmed 
or  modified  by  a  consideration  of  the  tables  of  results 
which  are  set  out  in  the  next  section. 

5.     Results  of  the  Experiments, 
(a)     Results  of  the  Preliminary  Tests. 

The  marks  for  the  four  preliminary  tests  were 
fairly  steady,  decidedly  so,  when  the  age  of  the  chil- 


THIRD   SEEIES   OF   EXPERIMENTS.  91 

dren  was  taken  into  consideration.  Very  few  of  the 
boys  made  any  violent  jumps,  and  there  was  a  gen- 
eral improvement  from  exercise  to  exercise. 

In  the  first  test  the  average  mark  was  11.1,  in  the 
second  12.3,  in  the  third  12.9,  and  in  the  fourth  13.1. 
The  correspondences  between  the  results  of  the  first, 
second,  third,  and  fourth  Preliminary  Tests  are 
shown  compendiously  in  the  following  table : 

Table  XV,  showing  the  correlation  between  the  results  of  the  four 
Preliminary  Tests. 

Marks  in 

the  four  No.         r-Average  Marks  in  Preliminary  Tests.-^ 

preliminary  of  First          Second          Third          Fourth 

tests.  boys.          test.  test.  test.  test. 

70  and  over 4  16.5  19.8  18.3  18.3 

60  to  70 8  14.8  14.4  16.5  17.3 

50  to  60 11  12.5  13.4  13.9  14.3 

40  to  50 15  10.4  11.3  12.1  11.9 

Below  40 12  6.7  7.9  8.9  9.0 

There  is  obviously  high  positive  correlation  be- 
tween the  results  of  the  successive  preliminary  tests. 
The  mental  functions  we  are  testing  appear  to  be 
working  very  steadily.  Exact  numerical  values  for 
the  coefficients  of  correlation  have  been  worked  out 
from  the  50  individual  cases  on  the  Pearson  formula. 
Between  the  results  of  Tests  1  and  2  the  correlation 
coefficient  is  +  .76  (probable  error  .04),  between 
Tests  2  and  3  is  +  .79  (probable  error  .03),  and  be- 
tween Tests  3  and  4  is  +  .80  (probable  error  .03). 
These  high  correlations  between  the  results  of  the 
successive  tests  give  us  reasonable  expectations  that 
a  division  into  two  equal  groups  may  be  satisfac- 
torily effected.  The  boys  were  divided  into  two 
equal  groups  containing  25  children  each.  The  fol- 
lowing table  will  indicate  how  the  division  was  made : 


92 


INDUCTIVE   VS.    DEDUCTIVE    METHODS. 


Table  XVI,  showing  the  Division  into  Two  Equal  Groups. 
Group  A. 


Name 
(Initials 

only). 
R.  D 


First. 

18 

A.  C 17 

L.   0 13 

W.  G.............  4 

G. "K.!!!".'.'.!.'.'.'.'.'.'  3 

Averages 11.1 

M.  V.'s 3.5 


-Marks  for  Preliminary  Tests. 
Second.        Third       Fourth. 


Name 
(Initials 
only). 
H    B  

First. 
17 

R    S  

14 

C.  L  

16 

S   B  

9 

A.  W... 

4 

22 

18 
16 


12.4 
2.9 

Group  B. 


18 
19 
18 

*6 
*7 

12.9 
2.5 


22 

16 
18 

12 

io 

13.3 
2.4 


-Marks  for  Preliminary  Tests.- 

Second.  Third.       Fourth. 

20  17  17 

19  19  18 

17  16  18 


Averages 11.1 

M.  V.'s.  .  2.7 


8 
*8 

12.3 

2.8 


11 
*6 

13.0 
2.7 


12.8 
3.3 


The  average  mark  per  boy  per  test  for  Group  A 
was  12.4  (mean  variation  2.6),  and  for  Group  B  was 
12.3  (mean  variation  2.6).  The  average  age  of  Group 
A  was  9  years  3  months,  and  of  Group  B  was  also 
9  years  3  months. 

(b)    Results  of  the  Tests  in  Immediate  and  Deferred 
Reproduction. 

It  now  remains  to  be  shown  which  of  the  two 
groups  was  the  more  successful  when  tested  on  pre- 
cisely what  they  had  been  taught  or  learnt. 


THIBD   SERIES   OF   EXPERIMENTS. 


93 


First 
deferred 
repro- 
duction. 
18.0 
3.0 
18.8 
3.4 

Second 
deferred 
repro- 
duction. 
18.1 
3.2 
19.4 
3.5 

First,  let  me  give  the  marks  of  the  two  groups  as 
wholes,  together  with  their  variability: 

Table  XVII,  shoiving  the  work  of  the  Inductive  and  Deductive 
Groups  compared,  in  the  Preliminary  Tests  and  in  the  Tests 
of  Immediate  and  Deferred  Reproduction  (positive  marks 
only). 

i Average  Marks.- 

First  Second 

For  all        imme-  imme- 

four          diate  diate 

preliminary   repro-  repro- 

tests.       duction.  duction. 

Inductive  group.. .  12.4            18.8  18.6 

M.  V.'s 2.6              2.6  2.5 

Deductive  group...  12.3            20.5  20.6 

M.  V.'s 2.6              3.4  4.1 

In  the  tests  for  deferred  reproduction,  it  will  be 
remembered,  negative  marks  were  given  as  well  as 
positive  marks.  The  marks  for  the  two  groups  are 
given  below  after  the  negative  marks  have  been  sub- 
tracted from  the  positive  marks : 

Table  XVIII,  showing  the  marks  (after  deduction)  for  the  Induct- 
ive and  Deductive  Groups  compared,  in  the  Preliminary  Tests, 
and  in  the  Tests  of  Deferred  Reproduction. 

Average  r- Average  Marks. -^ 

mark  First  Second 

for  four  pre-  deferred  deferred 

liminary  tests.  reproduction.  reproduction. 

Inductive  group 12.4  17.7  17.8 

M.  V.'s 2.6  3.2  3.4 

Deductive  group 12.3  18.6  19.1 

M.  V.'s 2.6  3.6  3.7 

There  seems  no  doubt  that,  when  the  tests  are  given 
on  precisely  the  subject-matter  which  has  been  learnt 
or  taught,  the  group  which  learnt  the  definitions  did 
better  work  than  that  which  was  taught  inductively, 
and  this  is  true  both  in  immediate  and  deferred  re- 
production, and  for  both  positive  and  negative 


94 


INDUCTIVE   VS.   DEDUCTIVE   METHODS. 


marks.  This  conclusion  must,  of  course,  be  drawn 
subject  to  the  age  and  mental  proficiency  of  the  pu- 
pils. It  now  remains  to  be  seen  whether  the  differ- 
ence between  the  groups  is  one  which  is  common  to 
the  more  proficient  as  well  as  to  the  less  proficient 
pupils : 

Table  XIX,  showing  the  marks  of  the  two  groups  compared,  section 
by  section,  in  the  Preliminary  Tests  and  in  the  Tests  of  Imme- 
diate and  Deferred  Reproduction  (positive  marks  only). 


Marks 

in  four  No. 

preliminary       of 

tests.          boys. 
70  and  over. . .  2 

60  to  70 4 

50  to  60 5 

40  to  50 8 

Below  40 6 

It  seems  clear  that  there  is  a  balance  of  advantage 
all  along  the  line  in  favor  of  the  group  which  learnt 
the  definitions,  so  far,  at  least,  as  the  positive  marks 
are  concerned.  It  now  remains  to  be  shown  whether 
this  is  also  true  when  the  negative  marks  are  de- 
ducted from  the  positive  marks : 

Table  XX,  showing  the  marks  of  the  two  groups  compared,  section 
by  section,  in  the  Preliminary  Tests  and  in  the  Tests  of  De- 
ferred Reproduction  (after  the  negative  marks  have  been  de- 
ducted). 

r-Inc 
Marks 

in  four  No. 

preliminary  of 

tests.  boys. 

70  and  over 2 

60  to  70 4 

50  to  60 5 

40  to  50 8 

Below  40 6 


•Inductive  Group.  * 

,  Deductive  Group.  * 

Average 

Average 

mark  of 

Average 

mark  of 

Average 

two  tests 

mark  of 

two  tests 

mark  of 

imme- 

two tests 

imme- 

two tests 

diate 

deferred 

No. 

diate 

deferred 

repro- 

repro- 

of 

repro- 

repro- 

duction. 

duction. 

boys. 

duction. 

duction. 

19.0 

19.2 

2 

24.0 

21.0 

21.2 

19.5 

4 

22.5 

20.9 

19.7 

19.2 

6 

22.9 

21.0 

18.0 

18.5 

7 

18.8 

18.3 

17.0 

14.8 

6 

17.8 

16.0 

tive  Group.—  ^ 

r-Deductive  Group.—  >> 

Average 

Average 

mark  in  two 

No. 

mark  in  two 

tests  deferred 

of 

tests  deferred 

reproduction. 

boys. 

reproduction. 

19.3 

2 

21.5 

19.1 

4 

20.5 

19.1 

6 

20.9 

18.4 

7 

18.4 

14.2 

6 

15.4 

THIRD    SERIES   OF   EXPERIMENTS.  95 

Again,  there  seems  a  decided  balance  of  advantage 
on  the  side  of  the  group  which  learnt  the  definitions 
deductively. 

(c)     Correlation  Between  the  Results  of  Immediate 
and  Deferred  Reproduction. 

It  would  seem  likely  from  the  tables  given  above 
that  the  tests  given  immediately  after  the  teaching 
and  learning  may  be  regarded  as  fairly  significant  of 
the  relative  position  of  the  two  groups  even  after 
considerable  time  has  elapsed — in  this  case  after  a 
month.  As  this  is  a  very  important  issue  for  experi- 
mental pedagogy,  it  may  be  well  to  subject  the  hy- 
pothesis to  further  determination.  The  following 
tables  will  show  in  a  general  way  how  far  the  sug- 
gestion may  be  taken  as  valid : 

Table  XXI,  showing  the  correlation  between  the  marks  obtained  in 
the  various  Tests  of  Reproduction  (positive  'marks  only). 

Inductive  Group. 

Marks  for  the 

first  test  of  No.          Average  Marks  per  Boy  in  the  Repro- 

immediate  of  ductive  Tests. 

reproduction.  boys.  First.       Second.       Third.       Fourth. 

Over  25 2  27.0  26.5  23.5  25.5 

20  to  25 3  22.3  19.7  20.7  20.7 

18  to  20 7  19.6  19.7  19.1  19.7 

16  to  18 6  17.7  18.1  16.0  15.7 

15  to  16 4  16.0  15.0  16.0  16.0 

15  and  under 3  14.3  15.3  16.0  14.3 

Deductive  Group. 
Marks  for  the 

first  test  of  No.          Average  Marks  per  Boy  in  the  Repro- 

immediate  of  ductive  Tests, 

reproduction.  boys.  First.       Second.       Third.       Fourth, 

Over  28 2  29.0  27.0  25.5  26.0 

25  to  28 3  26.7  26.0  22.3  22.7 

21  to  25 4  23.5  23.2  22.2  23.2 

17  to  21 8  19.7  20.4  18.5  19.7 

16  to  17 4  17.0  18.7  16.2  15.5 

16  and  under 4  13.7  13.2  12.5  12.7 


96  INDUCTIVE   VS.    DEDUCTIVE    METHODS. 

It  is  obvious  that  considerable  positive  correlation 
exists  between  the  results  of  the  successive  exercises. 
A  more  precise  determination  may,  of  course,  be 
made  by  means  of  a  correlation  coefficient.  Worked 
out  by  the  standard  formula  from  the  individual 
cases,  the  following  are  the  coefficients :  For  the  In- 
ductive Group  the  results  of  the  first  Test  of  Eepro- 
duction  correlate  with  those  of  the  second  to  the 
extent  of  +  .78,  the  second  with  the  third  to  the  ex- 
tent of  +  -57,  and  the  third  with  the  fourth  to  the 
extent  of  +  .85.  For  the  Deductive  Group  the  corre- 
lation coefficients  are :  first  and  second  tests,  +  .86 ; 
second  and  third  tests,  +  .68 ;  third  and  fourth  tests, 
+  .94. 

It  is  obvious  that  tests  given  immediately  may  be 
fairly  regarded  as  indicative  of  what  will  happen 
later  on,  at  least  in  such  exercises  as  these,  when  we 
are  making  comparison  of  one  group  with  another. 

(d)     Results  of  the  Test  on  Neiv  Material. 

We  have  seen  already,  when  the  tests  required  an 
exact  reproduction  of  what  had  been  learnt  or  taught, 
that  the  children  in  this  school  who  learnt  deduc- 
tively, like  the  Standard  V  children  of  the  girls' 
school  in  the  experiment  first  described,  did  better 
work  than  the  group  taught  inductively.  But  in  both 
the  experiments  previously  described  it  was  found, 
when  the  test  given  was  on  new  material,  that  the 
children  taught  inductively  did  better  work  than 
those  who  learnt  their  definitions.  Is  this  advantage 
also  to  be  found  on  the  side  of  the  inductive  group 
in  this  school  I  These  children  are  younger  and  are 
less  proficient  mentally,  according  to  school  grading, 


THIRD   SERIES   OF   EXPERIMENTS.  97 

than  either  of  the  girls'  classes  whose  work  has  been 
described.  Moreover,  they  are  boys,  not  girls.  Are 
these  variations  in  conditions  such  as  to  produce  a 
difference  in  the  results  ?  It  will  further  be  remem- 
bered that  the  inductive  group,  in  this  case,  was 
taught  by  its  own  teacher,  and  not  by  me,  so  that 
any  intensity  of  impression  due  to  personal  novelty 
was  thereby  eliminated. 

Perhaps  I  may  be  pardoned  for  a  sentence  of  ap- 
parent digression.  I  hold  it  extremely  important  for 
the  science  of  experimental  pedagogy  that  no  result 
should  be  taken  as  valid  for  general  application  un- 
less the  use  of  it  is  justified  by  its  success  in  the  hands 
of  the  usual  teachers  of  the  school.  Its  success  in 
the  hands  of  the  specialist  or  other  exceptional  per- 
son is  quite  insufficient  to  recommend  it  for  general 
adoption.  Let  us,  then,  see  what  the  results  were 
when  the  whole  experiment  was  conducted  by  the 
teachers  themselves.  I  shall  show  the  work  of  the 
two  groups  compared  both  in  the  Preliminary  Tests 
and  in  the  Test  of  Application  to  New  Material. 
First,  let  me  give  the  results  of  the  two  groups  as 
wholes : 

Table  XXII,  showing  the  ivork  of  the  tivo  groups  compared,  in  the 
Preliminary  Tests  and  in  the  Test  on  New  Material. 

Average  Mark  for  New 

Average  mark  Material. 

for  four  Positive  Marks  after 

preliminary  tests.  marks.  deduction. 

Inductive  group 12.4  16.3  15.6 

M.  V.'s 2.6  3.5  4.0 

Deductive  group 12.3  15.7  14.9 

M.  V.'s 2.6  5.8  5.2 

Again  we  find,  notwithstanding  the  decidedly  su- 
perior acquisition  of  the  material  studied  (see  Table 


98  INDUCTIVE   VS.    DEDUCTIVE   METHODS. 

XX)  on  the  part  of  the  deductive  group,  that  they  are 
inferior  to  the  other  in  their  power  to  attack  new 
material  of  an  analogous  nature.  Four  boys  in  the 
deductive  group  completely  failed  to  make  a  reason- 
able application  of  their  old  knowledge,  obtaining 
only  6,  7,  2  and  4  marks,  respectively,  whilst  only  one 
boy  in  the  inductive  group  failed  to  do  so,  and  he 
obtained  8  marks. 

Let  us  now  see,  as  we  have  in  previous  cases,  how 
far  this  difference  is  to  be  found  for  the  weaker  as 
well  as  for  the  abler  children  of  each  group : 

TaUe  XXIII,  showing  the  work  of  the  two  groups  compared,  sec- 
tion by  section,  in  the  Preliminary  Tests  and  in  the  Test  of 
Application  to  New  Material  (positive  marks,  and  positive 
marks  after  the  deduction  of  the  negative  marks). 

Group  Inductively  Taught. 

Marks  for  Application  to 

Marks  New  Material, 

in  four  No. 


preliminary                   of  Positive  After 

tests.                      boys.  only.  deduction. 

70  and  over 2  18.5  18.0 

60  to  70 4  18.3  17.0 

50  to  60 5  17.8  17.6 

40  to  50 8  15.5  15.0 

Below  40 6  14.0  13.2 

Group  Deductively  Taught. 

Marks  for  Application  to 

Marks  New  Material, 

in  four  No. 


preliminary                   of  Positive  After 

tests.                      boys.  only.  deduction. 

70  and  over 2  18.0  16.5 

60  to  70 4  15.0  14.2 

50  to  60 6  17.8  17.2 

40  to  50 7  14.0  13.3 

Below   40 6  15.3  14.6 

Except  in  the  case  of  the  least  proficient  section  of 
boys  at  the  bottom  of  each  group,  there  seems  to  be 


THIRD   SERIES   OF   EXPERIMENTS.  99 

an  advantage  all  along  the  line  in  favor  of  the  in- 
ductive group.  When,  therefore,  the  tests  are  tests 
of  the  application  of  knowledge  rather  than  an  exact 
reproduction  of  it,  we  are,  perhaps,  entitled,  on  the 
whole,  to  conclude  that  inductive  methods  are  the 
better.  It  may  be  noted  that  whilst  the  marks  for  the 
inductive  group  proceed  regularly  downwards  from 
the  highest  section  to  the  lowest,  those  for  the  corre- 
sponding sections  of  the  deductive  group  do  not.  The 
variability  for  this  group  is  disproportionally  high, 
due,  doubtless,  to  the  psychological  fact  that  for 
some  children  of  this  age  the  step  from  knowledge 
to  the  application  of  it  is  a  very  considerable  one; 
whereas,  of  course,  the  children  of  the  other  group 
had  been  through  a  process  of  applicable  method 
when  they  had  received  their  inductive  lesson.  The 
variability  of  the  work  is,  however,  decidedly  high, 
and  the  difference  between  the  means  of  the  work  of 
the  two  groups  is  very  small;  and  did  this  experi- 
ment stand  alone,  I  should  hesitate  before  putting 
much  confidence  in  the  conclusion  which  I  have  indi- 
cated above.  But  its  consilience  with  the  previous 
results  lends  strength  to  the  conclusion,  especially 
when  the  differences  in  the  conditions  under  which 
it  was  obtained  are  taken  into  account. 


' 


VII.    FOUETH   SERIES   OF   EXPERIMENTS. 
1.    General  Plan. 

In  the  experiment  now  to  be  described,  just  as  in 
those  previously  recounted,  the  work  was  done  with 
all  the  children  of  one  class,  under  one  teacher,  with 
the  same  curriculum  of  study,  and  working  according 
to  the  same  time-table  of  instruction.  The  experi- 
ment was  carried  out  in  a  municipal  higher  grade 
school  for  boys,  an  elementary  school  situated  in  a 
somewhat  mixed  neighborhood.  The  class  chosen 
for  the  experiment  was  the  First  Class  in  the  school, 
containing  35  boys,  graded  as  Ex.  VII  on  the  English 
standard  system  of  school  grading,  of  an  average  age 
of  approximately  IS1/^  years.* 

The  teacher  of  the  class  had  a  theoretical  acquaint- 
ance with  psychological  work,  and  had  already  car- 
ried out  some  observations  in  educational  psychol- 
ogy. He  was,  especially,  capable  of  temporary  dis- 
sociation between  the  pedagogic  and  psychologic 
attitudes — a  necessary  capacity  in  an  experimenter. 
Beyond  this,  he  was  a  first-rate  teacher  who  varied 
his  methods  according  to  the  subject-matter  with 
which  he  was  dealing. 

As  in  previous  cases,  the  class  was  divided  into 
two  equal  groups  on  the  results  of  tests  in  spontane- 

*In  America  this  would  constitute  Grade  VIII,  or  rather,  per- 
haps, the  First  Year  of  High  School. 

100 


FOUKTH   SERIES   OF   EXPERIMENTS.  101 

ous  definition,  but  the  test  on  which  the  division  was 
effected  was  not  the  same  as  that  used  for  the  pur- 
pose in  the  former  tests.  But,  as  before,  there  were 
tests  of  immediate  and  tests  of  deferred  reproduc- 
tion, and  a  test  of  application  to  new  material  of  an 
analogous  kind.  Further  relevant  conditions  will 
be  given  in  the  details  which  follow. 

2.     The  Preliminary  Tests  and  the  Method  of 
Marking. 

The  first  test  in  this  series  was  the  spontaneous 
definition  of  squares,  triangles,  oblongs,  and  diam- 
eters of  circles,  which  were  drawn  in  the  way  already 
indicated,  and  the  questions  (with  which  by  now  the 
reader  will  be  quite  familiar) :  "What  is  a  square?" 
etc.,  were  set  for  written  answers.  The  papers  were 
marked  on  the  system  of  units  which  has  already 
been  described,  and  an  average  mark  was  gained  of 
19.1  out  of  a  possible  maximum  of  30.  This,  as  might 
have  been  expected,  was  by  far  the  highest  mark  that 
had  been  obtained  by  any  class  doing  this  test.  It 
was  not  proposed  to  use  this  test  on  squares,  tri- 
angles, etc.,  for  the  purpose  of  dividing  the  class,  but 
it  served  a  useful  purpose  as  a  preparatory  exercise. 

Next  week  the  teacher  taught  all  the  children  of 
the  class  how  to  arrive  at  the  definitions  inductively 
in  the  way  that  I  have  described  in  the  first  series  of 
experiments  (p.  33).  This  lesson  also  rendered  val- 
uable service.  It  gave  full  opportunity  to  all  to  un- 
derstand quite  clearly  what  they  had  to  do  when  they 
were  set  to  attack  the  preliminary  test  on  which  the 
class  was  to  be  divided. 

The  questions  used  for  the  preliminary  test  were 


102  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

the  same  as  those  which,  in  the  previous  schools,  had 
been  used  as  a  test  of  the  power  of  application  to  new 
analogous  material.  In  one  sense  it  is,  of  course,  in 
this  case,  also  a  test  of  application  to  new  material, 
for  one  inductive  lesson  on  the  square,  etc.,  had 
already  been  given.  We  may,  indeed,  look  upon  our 
division  into  two  '  equal  groups '  in  this  case  as  being 
effected  during  the  course  of  a  series  of  lessons  in- 
stead of  at  the  very  beginning  of  it. 

The  questions  were:  "What  is  a  rhombus?" 
4  *  What  is  a  trapezium?"  "What  is  a  rhomboid?" 
and  "What  is  a  diagonal  of  a  square?"  The  an- 
swers were  marked  on  the  system  of  units  which  has 
already  been  described.* 

One  or  two  of  the  papers  worked  may  be  of  service 
in  enabling  an  experienced  teacher  to  gauge  the  men- 
tal level  of  the  boys  taking  these  tests. 

Edward  S ,  aged  14  years  8  months,  wrote : 

1.  A  rhombus  is  a  figure,  it  has  4  equal  straight  lines,  it  has 
angles,  there  are  4,  2  equal  large  ones  and  2  equal  small  ones. 

2.  A  trapezium  is  a  figure,  it  has  4  straight  lines  of  different 
lengths,  it  has  angles,  there  are  4,  all  of  different  sizes,     any  shape. 

3.  A  rhomboid  is  a  figure,  it  is  enclosed  by  4  straight  lines,  2 
equal  long  ones,  two  equal  short  ones.     It  has  4  angles,  2  equal 
small  ones  and  two  equal  large  ones. 

4.  A  diagonal  is  a  straight  line  going  from  one  corner  to  the 
other  of  a  square  terminating  at  both  ends  dividing  the  square  into 
2  triangles. 

This  is  not  the  best  paper ;  there  are  four  boys  in 
the  class  who  get  higher  marks,  but  it  is  obvious  that 
we  are  here  dealing  with  a  very  different  mental 
level,  geometrically,  from  those  at  which  we  have 
previously  worked.  With  the  table  of  units  at  hand, 


*The  reader  is  recommended  to  turn  to  page  41  for  the  list  of 
correct  units  of  description. 


FOURTH    SERIES   OF    EXPERIMENTS.  103 

it  is  quite  easy  to  mark  this  paper.  The  only  diffi- 
culty occurs  in  the  case  of  the  last  definition,  in  which 
the  qualification  opposite  is  omitted  when  the  corners 
of  the  square  are  mentioned.  It  is  held,  however, 
that  the  statement  "dividing  the  square  into  2  tri- 
angles'' is  equivalent  to  the  limitation  of  from  one 
corner  to  the  l opposite'  corner.  A  total  of  37  posi- 
tive marks  was  gained — 11  for  the  definition  of 
rhombus,  8  for  the  definition  of  trapezium,  14  for  the 
definition  of  rhomboid,  and  4  for  the  definition  of  a 
diagonal  of  a  square.  There  are  no  'bad  errors.' 
Charles  B ,  aged  13  years  9  months,  wrote : 

1.  A  rhombus  is  a  figure  or  drawing  consisting  of  4  straight 
lines.    All  the  lines  are  the  same  length  and  the  two  lines  opposite 
one  another  are  parrallel  to  one  another.    It  has  4  corners  and  four 
equal  angles. 

2.  A  trapezium  is  a  drawing.    It  has  4  straight  lines,  4  angles, 
all  the  lines  are  of  different  lengths.    It  has  two  long  and  two  short 
sides.    The  angles  are  all  different. 

3.  A  rhomboid  is  a  drawing  consisting  of  4  lines  which   are 
straight.    It  has  2  lone:  and  2  short.    The  2  short  are  parrallel  to 
one  another,  and  the  2  long  are  parrallel  to  one  another.     It  has 
4  corners  and  four  angles.     The  2  long  sides  are  the  same  length 
and  the  2  short  are  the  same  length.    All  the  angles  are  not  equal. 

4.  A  diagonal  is  a  line,  must  be  straight.    It  is  drawn  from  one 
corner  to  the  one  opposite.     It  passes  through  the  centre  of  the 
figure.     It  does   not  go  outside  the   figure.     It  must  touch  the 
corners. 

This  also  is  a  good  paper,  gaining  one  mark  more 
than  the  average  for  the  whole  class.  The  marks  are 
quite  easily  given.  The  definition  of  rhombus  gains 
9  positive  marks.  There  is  one  'bad  error' — the  four 
angles  are  not  equal;  but  throughout  this  experiment 
we  worked  with  positive  marks  only.  The  definition 
of  trapezium  gains  all  the  positive  marks  possible 
on  our  scale  of  marking,  namely,  8.  It  is  interesting 
to  note  that  the  term  *  corners'  is  not  in  C.  B.'s  mind 
synonymous  with  'angles.'  In  every  one  of  his  defi- 


104  INDUCTIVE   VS.   DEDUCTIVE   METHODS, 

nitions  in  which  the  confusion  can  occur  he  makes 
the  same  duplication,  but  these  duplications  are  not 
' bad  errors'  according  to  our  system  of  marking. 
The  definition  of  rhomboid  receives  12  positive 
marks,  and  the  definition  of  a  diagonal  of  a  square 
receives  full  marks,  namely,  4.  The  total  marks  for 
this  paper  amount  to  33.  It  is  scarcely  necessary  to 
multiply  examples;  sufficient  have  been  given  to 
show  how  much  more  competent,  geometrically 
speaking,  these  boys  are  than  those  with  whom  we 
worked  in  the  previous  boys '  school.  I  turn  now  to 
the  chronology  of  the  whole  of  the  experiment. 

3.     Chronology  of  the  Experiment. 

The  first  test  in  this  series  was  given  on  Friday 
at  9.30  A.  M.,  September  29,  immediately  after  Scrip- 
ture lesson.  In  this  test  the  boys  were  asked,  un- 
taught and  unaided,  to  define  square,  triangle,  etc. 
Exactly  one  week  later  all  the  boys  in  the  class  were 
taught  inductively  how  to  arrive  at  the  definitions 
of  square,  triangle,  etc.  On  Tuesday,  October  17, 
at  11  o'clock,  immediately  after  recreation,  the  test 
used  as  a  Preliminary  Test  in  this  school,  "What  is 
a  rhombus  ? ' '  etc.,  was  given,  on  the  results  of  which 
the  class  was  divided  into  two  equal  groups.  In  this 
experiment  one  test  only  was  given  for  purposes  of 
division.  It  was  hoped  that  the  preparatory  work 
with  the  squares,  triangles,  etc.,  together  with  the 
greater  age  and  proficiency  of  the  children,  would 
result  in  the  necessary  steadiness,  and  that  the  boys' 
variability  would  be  so  small  that  one  test  would 
suffice. 

On  Thursday  afternoon,  October  19,  from  2.30  to 
2.50,  one  of  the^ groups  was  taught  inductively  how  to 


FOUKTH   SEBIES   OF   EXPEBIMENTS.  105 

arrive  at  the  definitions  of  rhombus,  trapezium, 
rhomboid,  and  diagonal  of  a  square.  Exactly  the 
same  method  was  followed  as  that  used  by  me  in  the 
first  and  second  experiments.  The  teacher  of  the 
class  had  heard  me  ' teach'  the  definitions,  so  there 
was  no  danger  that  he  would  vary  the  method  essen- 
tially ;  but  a  minor  variant  was  employed.  He  jotted 
down  on  the  blackboard  (which  I  did  not),  in  an  ab- 
breviated form,  the  i units'  of  description  as  the  boys 
supplied  them.  His  argument  for  the  variation  was 
that  the  boys  who  were  going  to  study  the  definitions 
deductively  would  have  visual  memories  of  verbal 
descriptions  to  help  them,  and  that  the  inductively 
taught  group  ought  also  to  have  some  visual  verbal 
memories  to  assist  them.  Whilst  the  boys  of  one 
group — Group  A — were  being  taught  the  definitions, 
the  other  group — Group  B — went  into  the  school  hall 
and  had  a  reading  lesson  under  a  student-teacher. 
From  2.55  to  3.15  the  boys  of  Group  A  went  into  the 
school  hall  and  took  the  reading  lesson,  whilst  Group 
B  came  back  to  their  own  teacher  and  studied  the 
definitions  of  rhombus,  etc.,  which  had  been  con- 
structed from  the  spontaneous  descriptions  of  the 
Preliminary  Test  and  had  been  already  written  in 
preparation  upon  a  blackboard,  with  the  appropriate 
drawings. 

Definitions  of  Rhombus,  Trapezium,  Rhomboid  and 

Diagonal  of  a  Square  in  the  Form  in  Which 

They  Were  Given  to  the  'Deductive' 

Group  to  Study. 

A  rhombus  is  a  figure  enclosed  by  4  equal  straight 
lines.  Two  sides  opposite  are  parallel,  and  the  other 
two  sides  opposite  are  parallel.  It  has  4  angles,  2 


106  INDUCTIVE   VS.   DEDUCTIVE    METHODS. 

large  and  2  small.  The  2  large  angles  are  equal  and 
opposite,  and  the  2  small  angles  are  also  equal  and 
opposite. 

A  trapezium  is  a  figure  enclosed  by  4  unequal 
straight  lines.  It  contains  4  unequal  angles. 

A  rhomboid  is  a  figure  enclosed  by  4  straight  lines, 
2  long  and  2  short.  Two  long  sides  are  equal,  oppo- 
site and  parallel,  and  the  two  short  sides  are  equal, 
opposite  and  parallel.  It  has  4  angles,  2  large  and  2 
small.  The  2  large  angles  are  equal  and  opposite, 
and  the  2  small  angles  are  equal  and  opposite. 

A  diagonal  of  a  square  is  a  straight  line  which 
starts  at  an  angle  and  passes  across  the  square  to  the 
opposite  angle. 

The  boys  were  told  to  study  the  definitions,  and 
they,  as  well  as  the  boys  of  the  inductive  group,  were 
made  aware  that  they  would  be  required  to  answer 
questions  on  them.  The  time  from  3.15  to  3.30  was 
spent  by  the  boys  of  both  groups  in  the  playground. 
At  3.30  all  the  boys  returned  to  their  classroom ;  the 
questions,  "What  is  a  rhombus!"  etc.,  which  had 
been  written  on  the  blackboard,  were  exposed  to 
view,  and  the  boys  wrote  the  answers.  There  was 
one  other  variant  from  the  method  which  I  had  used 
myself,  for  the  drawings  of  the  figures  were  placed 
before  the  boys  whilst  they  were  answering  the  ques- 
tions in  their  tests  of  reproduction. 

Exactly  one  week  later,  on  Thursday,  October  26, 
at  3.30  P.  M.,  the  boys  of  both  groups  worked  a  test  in 
deferred  reproduction,  following  immediately  upon 
the  recreation  interval,  as  in  the  test  of  immediate 
reproduction. 

Two  weeks  after  this  test,  at  10  o'clock  in  the 
morning,  on  Wednesday,  November  8,  following  two 


FOURTH   SERIES   OF   EXPERIMENTS.  107 

short  lessons  on  Scripture  and  French  reading,  the 
test  of  Application  to  New  Material  was  given. 

Perhaps  a  summarized  note  showing  the  main 
chronological  issues  involving  differences  from  other 
experiments  may  be  of  service.  First,  both  groups 
had  inductive  teaching,  as  well  as  inductive  practice, 
before  the  Preliminary  Test.  There  was  one  pre- 
liminary test,  and  only  one.  The  test  of  deferred  re- 
production was  given  one  week  after  the  test  of  im- 
mediate reproduction.  The  test  of  application  to 
new  material  was  given  three  weeks  after  the  teach- 
ing and  learning  which  we  were  relying  on  to  differ- 
entiate the  groups,  and  two  weeks  after  the  test  of 
deferred  reproduction. 

4.    The  Tests  of  Immediate  and  Deferred  Repro- 
duction. 

In  these  tests  all  the  boys  in  the  class  answered  in 
writing  the  following  questions:  "What  is  a  rhom- 
bus?" etc.  The  questions  were  written  on  the  black- 
board, and  the  drawings  of  the  rhombus  and  other 
figures  were  again  shown  to  the  boys.  I  have  already 
pointed  out  that  this  was  a  variation  on  the  method 
previously  adopted. 

5.     The  Test  of  Application  to  Neiv  Material. 

Drawings  of  hexagons,  pentagons,  tangents  and 
quadrilaterals  (similar  figures),  with  the  names  ap- 
pended, were  shown  to  the  boys  thus : 


108  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

HEXAGONS. 


PENTAGONS. 


TANGENTS    TO   CIRCLES. 
(The  tangents  are  drawn  in  dots.) 


The  sides  of  LMNO  were  each  1%  times  the  corresponding  sides 
of  ABCD,  so  that  no  easily  recognizable  ratio  should  appear.  The 
figures  were  drawn  so  that  CD  and  NO  were  not  quite  in  the  same 
straight  line. 

In  the  diagrams  actually  used  the  tangents  were  continuous  lines 
drawn  in  red. 


FOUBTH   SEEIES   OF   EXPERIMENTS.  109 

Then  the  following  questions  were  written  on  a 
blackboard  and  the  boys  required  to  answer  them  in 
writing : 

1.  "What  is  a  hexagon?" 

2.  "What  is  a  pentagon!" 

3.  "What  is  a  tangent  to  a  cicle!" 

4.  "In    how    many    ways    is    ABCD    like 

LMNO?" 

The  boys  were  allowed,  nay  encouraged,  to  give 
thought  and  time  to  their  answers.  It  will,  doubtless, 
be  remembered  that  no  time  limits  were  imposed  in 
any  of  the  tests  and  exercises  in  these  experiments. 
The  system  of  marking  the  papers  could,  no  doubt, 
be  inferred  by  analogy  from  the  units  of  correct  de- 
scription which  the  boys  and  girls  have  given  in  other 
cases  and  which  we  have  adopted.  But  it  is  unneces- 
sary for  us  to  infer  what  our  units  ought  to  be ;  they 
emerge  quite  clearly  from  a  consideration  of  the  pa- 
pers actually  worked. 

Let  me  give  one  or  two  by  way  of  illustration  be- 
fore listing  the  units  on  which  the  boys '  papers  were 
marked. 

Frederic  R ,  aged  13  years  11  months,  who 

worked  in  the  deductive  group,  wrote : 

1.  A  hexagon  is  a  figure  enclosed  by  six  equal  straight  lines.    It 
has  six  angles,  all  equal.    The  two  opposite  sides  are  parallel  in 
the  three  cases. 

2.  A  pentagon  is  a  figure  enclosed  by  five  equal  straight  lines. 
It  has  five  angles  all  equal.    None  of  the  sides  are  parallel  to  each 
other. 

3.  A  tangent  to  a  circle  is  a  straight  line  any  length,  which  must 
touch  the  side  of  the  circle  anywhere,  but  must  not  cut  it. 

4.  The  first  thing  why  ABCD  differs  from  LMNO  is  its  size. 
The  4  angles  are  the  same  in  both  figures.    The  4  straight  lines  are 
the  same  only  in  proportion,  LMNO  is  about  half  the  size  again 
as  ABCD.    M.N.O.  angles  are  the  same  as  B.D.C.  only  the  sides  are 


110  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

different  lengths.  A.  angle  is  exactly  the  same  as  L.  angle.  Both 
the  figures  are  exactly  the  same  shape.  The  only  thing  why  one 
is  different  from  the  other  is  in  size. 

Even  without  a  list  of  units  of  correct  description 
it  is  not  difficult  to  assess  this  paper.  The  definition 
of  hexagon  receives  a  mark  for  '  figure, '  four  marks 
for  "six  equal  straight  lines,"  three  marks  for  "six 
angles  equal,"  and  six  marks  for  noting  that  there 
were  three  pairs  of  opposite  sides,  and  that  three 
pairs  were  parallel.  F.  E.  thus  obtains  a  total  of  14 
marks  for  his  definition  of  hexagon.  The  definition 
of  pentagon  receives  eight  marks — one  for  *  figure,' 
four  for  "five  equal  straight  lines,"  and  three  for 
"five  angles  equal."  The  definition  of  tangent  re- 
ceives three  marks — one  for  Mine,'  one  for  '  straight,' 
and  one  for  "touching  the  side  of  the  circle."  A 
boy's  conception  of  touching  would  be  satisfied  if  the 
line  impinged  upon  the  circumference  of  the  circle 
in  such  a  way  that,  if  produced,  it  would  cut  the  cir- 
cumference. Consequently  it  is  necessary  to  add  the 
limitation  'if  produced,  will  not  cut  the  circle.'  The 
fourth  answer  is  a  good  one,  but  it  is  unfortunate  that 
the  boy  is  bothered  by  the  notion  that  he  has  to  find 
differences,  which  every  now  and  again  intervene 
among  the  similarities.  He  calls  ABCD  and  LMNO 
both  '  figures,'  for  which  he  receives  a  mark;  for  '4 
angles '  he  receives  two  more ;  for  noting  that  the  four 
angles  are  equal,  each  to  each,  in  the  two  figures,  he 
receives  four  marks;  for  "4  straight  sides"  three 
more,  and  for  the  similar  proportionality  of  the  sides 
he  obtains  four  more.  Finally,  he  notes  that  the  fig- 
ures are  alike  in  shape,  for  which  he  receives  a  mark. 
F.  E.  's  total  mark  for  this  answer  is  15,  and  his  mark 
for  his  whole  paper  40.  His  marks  were  38  for  his 


FOURTH    SERIES   OF   EXPERIMENTS.  Ill 

preliminary  test,  49  for  immediate  reproduction 
after  teaching,  49  for  deferred  reproduction,  and  40 
— the  present  mark — for  application  to  new  mate- 
rial. If  these  are  compared  with  the  average  marks 
given  later,  it  will  be  seen  that  he  is  five  or  six  marks 
ahead  of  the  average  throughout  the  entire  series. 
I  will  give  one  or  more  worked  papers  before  setting 
out  the  units  of  correct  definition  which  were  ac- 
cepted as  the  basis  of  marking. 

Eobert  S ,  aged  14  years,  who  worked  in  the 

inductive  group,  wrote : 

1.  A  hexagon  is  a  figure  enclosed  by  six,  straight,  equal,  sides. 
The  opposite  sides  are  equal  and  parallel.     One  side  is  exactly 
balanced  by  the  opposite  one.     It  has  six  angles,  which  are  all 
equal.    Three  are  on  one  side  and  three  on  the  other. 

2.  A  pentagon  is  a  figure  enclosed  by  five  straight  sides.    They 
may  be  equal  or  unequal.    No  sides  are  opposite  and  no  sides  are 
parallel.    It  have  five  angles.    They  may  be  equal  or  unequal. 

3.  A  tangent  to  a  circle  is  a  straight  line.    It  may  be  drawn  at 
any  angle.    It  must  touch  the  circle  but  not  cut  it. 

4.  Both  have  five  sides.     The  base  in  each  case  is  horizontal. 
They  have  five  angles  each.    The  angles  are  the  same  number  of 
degress  in  each  case.    There  are  two  large  ones  and  two  small  ones. 
The  two  large  ones  are  formed  by  the  base  and  sides  and  the  two 
small  ones  from  the  top  and  sides.    The  smallest  angle  is  A  in  1 
and  correspondonds  with  L  in  2.    The  largest  C  in  1  and  N  in  2. 

The  definition  of  hexagon  obtains  14  marks — one 
for  *  figure/  four  for  "six  straight  equal  sides;"  six 
for  noting  that  there  are  three  pairs  of  opposite 
sides,  and  that  they  are  parallel,  each  to  each,  and 
three  marks  for  "six  equal  angles."  The  definition 
of  pentagon  obtains  six  positive  marks — one  for  "fig- 
ure," three  for  "five  straight  sides,"  and  two  for 
"five  angles."  The  statement  'no  sides  are  opposite 


112  INDUCTIVE   VS.    DEDUCTIVE   METHODS. 

and  no  sides  are  parallel7  is  held  to  be  of  too  negative 
a  nature  for  inclusion  within  the  definition.  To  say 
that  the  sides  and  the  angles  may  be  equal  or  unequal 
would  be  accounted  'bad  errors,'  though,  as  I  have 
said  before,  we  did  not  tabulate  the  'bad  errors'  in 
this  fourth  experiment.  The  definition  of  tangent 
receives  three  marks — one  for  'straight,'  one  for 
'line,'  and  one  for  'touch  the  circle.'  "It  may  be 
drawn  at  any  angle"  is  too  vague  to  be  regarded  as 
either  positive  or  negative.  The  point  is  missed  that 
the  tangent,  if  produced,  will  not  cut  the  circle.  In 
the  fourth  answer  there  are  two  curious  errors.  The 
figures  have  4  sides  and  4  angles,  and  not  5,  as  E.  S. 
says.  He  obtains  marks  for  mentioning  'sides'  and 
'angles'  as  pertaining  to  both.  Nearly  all  the  rest 
of  the  answer  is  occupied  with  the  equality  of  the 
angles  each  to  each,  for  which  4  marks  are  obtained. 
One  mark  is  gained  for  noting  that  the  base  lines  in 
each  case  are  horizontal ;  that  is  regarded  as  equiva- 
lent to  noting  that  their  inclination  is  the  same.  This 
marking  yields  a  total  of  30  positive  marks,  with  4 
'bad  errors.'  I  give  this  paper  because  I  wish  to 
make  it  quite  clear  that  boys  inductively  taught 
could  quite  well  make  'howlers'  as  well  as  boys  de- 
ductively taught,  though  these  boys,  in  both  groups, 
make  extremely  few.  E.  S.'s  other  marks  were  26, 
42  and  43 ;  in  all  cases,  except  that  of  the  Deferred 
Eeproduction  Test,  well  below  the  average.  Prob- 
ably the  perusal  of  the  papers  given  above  may  make 
clearer  the  usefulness  of  the  units  of  correct  defini- 
tion which  are  now  appended. 


FOUBTH    SEBIES   OF    EXPERIMENTS.  113 

Units  of  Correct  Description  or  Definition  of 
Hexagon,  etc. 

1.  A  hexagon  is  a  figure. 
It  has  sides  or  lines. 

It  has  straight  sides  or  lines. 
It  has  equal  sides. 
It  has  six  equal  sides. 
Two  sides  are  opposite. 
Two  other  sides  are  opposite. 
And  the  two  other  sides  are  opposite. 
Two  opposite  sides  are  parallel. 
Other  two  opposite  sides  are  parallel. 
And  the  other  two  opposite  sides  are  parallel. 
It  has  angles. 

Its  angles  are  six  in  number. 
And  they  are  equal. 
Its  angles  are  greater  than  right  angles. 
(A  total  of  15  points.) 

2.  A  pentagon  is  a  figure. 
It  has  sides  or  lines. 
Its  sides  are  straight. 
The  sides  are  equal. 
There  are  five  sides. 

It  has  angles. 

Its  angles  are  five  in  number. 
The  angles  are  equal. 
And  they  are  greater  than  right  angles. 
(A  total  of  9  points.) 

3.  A  tangent  to  a  circle  is  a  line. 
It  is  a  straight  line. 

The  line  touches  the  circle. 
And,  if  produced,  does  not  cut  it. 
(A  total  of  4  points.) 


114  INDUCTIVE  VS.   DEDUCTIVE   METHODS. 

4.    ABCD  and  LMNO  are  both  figures. 
They  both  have  sides. 
They  both  have  straight  sides. 
Their  sides  are  in  both  cases  unequal. 
And  they  are  4  in  number  in  both  figures. 
They  both  have  angles. 
Their  angles  are  4  in  number. 
And  are  in  both  cases  unequal  angles. 
BA  is  the  same  fraction  of  LM. 
As  BC  is  of  MN. 
As  CD  is  of  NO. 
As  AD  is  of  LO. 

BA  has  the  same  slant  or  is  parallel  to  LM. 
BC  has  the  same  slant  or  is  parallel  to  MN. 
CD  is  parallel  to  NO. 
And  AD  is  parallel  to  LO. 
The  angle  at  A  equals  the  angle  at  L. 
The  angle  at  B  equals  the  angle  at  M. 
The  angle  at  C  equals  the  angle  at  N. 
The  angle  at  D  equals  the  angle  at  0. 
The  figures  have  the  same  shape. 
(A  total  of  21  points.) 

It  is,  of  course,  not  urged  that  the  common  proper- 
ties of  the  figures  have  been  exhaustively  enumer- 
ated, but  only  that  the  units  of  correct  description 
are  such  as  are  actually  used  by  boys  and  are  service- 
able for  marking  papers  in  such  experiments  as 
these. 

6.    Results. 

First,  let  me  give  the  coefficients  of  correlation  be- 
tween the  results  for  the  various  exercises  in  so  far 
as  they  may  be  of  value.  The  marks  for  the  Prelimi- 


FOURTH   SERIES   OF   EXPERIMENTS.  115 

nary  Test  in  the  A  Group  were  very  closely  corre- 
lated with  those  of  the  B  Group ;  the  boys  were  most 
successfully  paired  in  the  two  groups,  from  the  best 
downwards  to  the  worst.  Worked  out  on  the  regular 
formula,  the  coefficient  of  correlation  amounted  to 
+  .98.  The  results  of  the  test  in  immediate  repro- 
duction correlated  with  that  of  deferred  reproduction 
to  the  extent  of  +  .752  in  the  inductive  group  and 
+  .777  in  the  deductive  group.  There  was  a  falling 
off  on  the  average  of  about  one  unit  in  the  marks. 
There  were  7  cases  out  of  34  in  which  the  mark  for 
deferred  reproduction  was  higher  than  for  immediate 
reproduction,  10  cases  in  which  it  was  the  same,  and 
17  cases  in  which  there  was  a  decline.  The  decline 
of  the  whole  class  was  from  an  average  mark  per  boy 
of  44.94  to  43.50,  with  mean  variations  approximat- 
ing to  4  in  both  cases,  and  a  correlation  coefficient 
between  the  results  of  immediate  and  deferred  re- 
production of  +  .78.  Though  the  difference  is  small, 
we  are  entitled  statistically  to  say  that  there  is  a 
general  tendency  to  decline,  since  the  difference  be- 
tween the  means  is  from  three  to  four  times  the 
probable  error  of  that  difference.  A  general  slight 
decline  seems,  therefore,  clear.  The  inductive  group 
falls  from  45.2  to  43.5;  the  deductive  from  44.7  to 
43.5.  But  the  fall  is  too  irregular  to  enable  us  to  con- 
clude that  there  is  any  greater  tendency  to  loss  on 
the  part  of  the  children  inductively  taught  than  of 
those  deductively  taught. 

Let  us  now  consider  the  results  of  the  test  on  new 
material.  It  is  clear  that  the  difference  between  the 
results  of  the  two  groups  is  very  small  in  this  school, 
though  it  favors  the  inductive  group,  as,  indeed,  is 
the  case  in  all  the  experiments.  But  the  variability 


116 


INDUCTIVE   VS.   DEDUCTIVE   METHODS. 


is  such  that  without  very  high  positive  correlation 
between  the  two  series  the  probable  error  of  the  dif- 
ference between  the  means  will  be  considerable. 

Now  let  me  give  the  average  results  in  gross,  treat- 
ing the  groups  as  wholes.  There  were  17  boys  in 
each  group : 

Table  XXIV,  showing  the  work  of  the  Inductive  and  Deductive 
groups  compared,  in  the  Preliminary  Test,  in  the  Tests  of  Im- 
mediate and  Deferred  Reproduction,  and  in  the  Test  of  Appli- 
cation to  Neiv  Material. 

Test  of 

immediate 

repro- 


Pre- 

liminary 

Inductive  Group:  test.  duction. 

Average  mark...  32.06  45.18 

M.   V 3.13  3.67 

Deductive  Group: 

Average  mark...  32.12  44.71 

M.   V 2.98  4.69 


Test  of 
deferred 

repro- 
duction. 

43.53 
3.97 


Test  on 

new 
material. 

35.65 
3.24 


43.47 
4.80 


35.00 
3.53 


The  boys  of  the  Inductive  Group  appear  to  hold 
the  advantage  throughout,  though  they  were  slightly 
weaker  in  the  Preliminary  Test.  A  closer  analysis 
of  the  results  is  given  in  the  next  table : 

Table  XXV,  shoioing  the  ivork  of  the  Inductive  and  Deductive 
Groups  compared,  section  by  section,  in  the  Preliminary  Test, 
the  Tests  of  Immediate  and  Deferred  Reproduction,  and  the 
Test  of  Application  to  Neiv  Material. 


Inductive  Group. 


Marks  in  No. 

preliminary  of 

test.  boys. 

Over  35 4 

30  to  35 6 

25  to  30 7 


Marks  in  No. 

preliminary  of 

test.  boys. 

Over  35 4 

30  to  35 6 

25  to  30 7 


Pre- 
liminary 
test. 
37.50 
32.50 
28.57 


-Average  Mark  per  Boy.- 


Immediate 
repro- 
duction. 
47.00 
47.00 
42.57 


Deferred 

repro- 
duction. 

45.50 

45.83 

40.42 


New 
material. 
37.50 
37.66 
32.85 


Deductive  Group. 


-Average  Mark  per  Boy.- 


Pre- 

Immediate 

Deferred 

liminary 
test. 

repro- 
duction. 

repro- 
duction. 

New 
material. 

37.50 

47.25 

47.75 

37.00 

32.33 

44.16 

43.16 

36.66 

28.85 

43.71 

41.28 

32.42 

FOUKTH   SEEIES   OF   EXPEBIMENTS.  117 

Only  in  one  test — that  of  application  to  new  mate- 
rial— does  there  appear  to  be  a  regular  sectional  ad- 
vantage on  the  side  of  the  inductive  group,  both  for 
the  weaker  as  well  as  for  the  stronger  boys.  In  both 
reproductive  tests  the  balance  of  advantage  shifts 
from  side  to  side. 

We  may  justifiably  conclude  that  the  results  of  this 
experiment,  having  regard  to  the  greater  age  and 
mental  ability  of  the  children,  are  consilient  with 
those  of  the  former  researches.  The  inductive 
method  has  shown  itself  the  better  when  application 
to  new  analogous  material  is  the  test  employed.  We 
are  unable  to  say  with  any  confidence  which  of  the 
two  groups  has  been  the  more  successful  in  immedi- 
ate and  deferred  reproduction.  The  average  results 
are  slightly  in  favor  of  the  inductive  group,  but  the 
balance  of  advantage  fluctuates  from  side  to  side,  and 
is  decidedly  uncertain.  But  this  is  the  fourth  case 
in  which  the  inductive  method  has  shown  itself  supe- 
rior in  application  to  new  material,  and  the  second 
case  in  which  the  inductive  method  has  equaled  the 
other,  even  for  purposes  of  reproduction.  In  both 
these  classes  there  had  been  much  previous  inductive 
teaching.  But  it  must  be  remembered  as  well  that 
the  class  of  much  younger  boys,  in  which  the  deduct- 
ive group  scored  heavily  in  reproductive  work,  were 
also  accustomed  to  much  inductive  work.  Age  ap- 
pears to  be  a  factor ;  perhaps  it  is  the  younger  chil- 
dren who  reproduce  better  on  a  deductive  and  memo- 
riter  method.  This  hypothesis  will  be  put  to  the  test 
in  the  last  of  this  series  of  experiments. 


VIII.     FIFTH  SERIES  OF  EXPERIMENTS. 

1.    General  Plan. 

Just  as  in  the  previous  experiment,  a  whole  class, 
working  under  one  teacher,  and  according  to  the 
same  syllabus  of  instruction,  with  the  same  time- 
table of  work,  was  divided  into  two  equal  groups  on 
the  results  of  a  test  in  the  definition  of  geometrical 
forms,  which  the  boys  attempted,  untaught  and  un- 
aided. Then  one  group  worked  inductively  and  the 
other  deductively.  Tests  were  given  immediately 
after  the  teaching  and  learning,  also  in  deferred  re- 
production a  week  later,  and  in  reproduction,  still 
further  deferred,  about  seven  weeks  after  the  first 
test  of  deferred  reproduction.  About  two  weeks 
after  the  teaching  and  learning  a  test  of  application 
to  new  material  was  given.  The  boys  who  did  the 
work  were  graded  as  Standards  VI,  a,  and  VII ;  their 
average  age  was  12  years  9%  months,  and  their 
teaching  generally  had  been  clear  and  efficient,  but 
had  tended  rather  towards  deductive  than  inductive 
methods.  The  school  was  situated  in  a  poor  neigh- 
borhood in  the  southeast  of  London,  and  the  average 
mental  ability  of  its  pupils  was  low  ;*  but  the  boys  of 
the  highest  class,  with  whom  the  experiment  was 
made,  were  by  no  means  without  ability ;  in  fact,  in 

*The  natural  mental  ability  of  the  pupils  of  this  school  was 
well  known  to  me  from  the  results  of  a  number  of  mental  tests 
which  had  been  applied  to  every  child  over  eight  years  of  age. 

119 


120  INDUCTIVE   VS.    DEDUCTIVE    METHODS. 

consequence  of  certain  exigencies  of  organization, 
the  class  contained  more  children  of  ability  than 
would  ordinarily  be  found  in  a  top  class  of  such  a 
size  in  a  school  of  this  social  type. 

2.    The  Preliminary  Tests  and  the  Method  of 
Marking. 

The  Preliminary  Test,  on  the  results  of  which  the 
boys  were  divided  into  two  equal  groups,  was  the 
same  as  that  used  in  the  experiment  just  described, 
which  took  place  in  the  higher  grade  school.  The 
teacher  had  already  used  the  questions:  "What  is  a 
square! "  "What  is  a  triangle?"  "What  is  an  ob- 
long?" and  "What  is  a  diameter  of  a  cicle?"  (with 
the  consideration  of  the  appropriate  drawings)  as  a 
kind  of  general  propaedeutic  to  the  experimental 
series,  and  the  boys  had  already  been  shown  in- 
ductively how  to  work  out  the  definitions  of  square, 
triangle,  oblong  and  diameter  just  as  they  had  in  the 
higher  grade  school. 

The  Preliminary  Test,  given  two  or  three  weeks 
later,  consisted  in  the  questions:  "What  is  a  rhom- 
bus?" "What  is  a  trapezium?"  "What  is  a  rhom- 
boid?" and  "What  is  a  diagonal  of  a  square?"  The 
appropriate  drawings  were  shown  and  the  boys  at- 
tempted to  answer  the  questions.  I  give  below  one 
or  two  of  their  worked  papers. 

William  L ,  aged  13  years  8  months,  wrote : 

1.  A  Rhombus  is  a  figure  with  all  side  equal  two  sides  slope  at 
60°  and  the  other  two  run  parallel. 

2.  A  Trapezium  is  a  figure  with  four  unequal  sides,  and  it  as  a 
right  angle  in  it. 

3.  A  Rhomboid  is  a  figure  with  two  long  sides  equal  and  two 
short  sides  equal,  but  none  of  the  corners  have  right  angles. 

4.  A  Diagonal  of  a  Square  is  the  distance  across  from  corner  to 
another  corner  which  slopes  at  45°. 


FIFTH    SERIES   OF    EXPERIMENTS.  121 

W.  L.,  in  his  first  definition,  gains  a  total  of  six 
marks.  His  second  definition  receives  four  marks. 
4 ' It  as  a  right  angle  in  it"  is  not  true  as  applied  to 
all  the  trapeziums;  it  is  a  'bad  error,'  but  there  are 
so  few  of  these  that  they  are  not  tabulated.  The 
definition  of  rhomboid  receives  eight  marks.  The 
negative  statement  that  there  are  no  right  angles, 
though  correct,  receives  no  mark,  as  we  could  hardly 
have  made  allowance  for  all  the  negative  statements 
which  may  be  truly  made  about  the  figures.  The  defi- 
nition of  diagonal  receives  two  marks  only — one  for 
' distance'  and  one  for  "from  corner  to  another 
corner."  W.  L.'s  paper  receives  a  total  of  20  posi- 
tive marks.  It  is  one  of  the  best  papers  worked  in 
the  class,  and  is  assessed  considerably  above  the  av- 
erage mark,  which  is  12.25  for  this  preliminary  test. 

Let  me  now  give  the  paper  of  a  boy  who  is  among 
those  toward  the  bottom  of  the  lists. 

Frank  B ,  aged  12  years  4  months,  wrote : 

1.  A  Rhombus  is  a  square  turned  in  shape. 

2.  A  Trapezium  is  a  figure  all  sides  unequal. 

3.  A  Rhomboid  is  an  oblong  with  the  two  smallest  perpendicular 
lines  slanting. 

4.  Diagonal  of  a  square  is  a  line  drawn  from  top  to  bottom  of 
the  corners. 

As  we  have  seen  in  former  cases  of  ' unintelligent' 
children,  the  similarities  between  these  figures  and 
those  which  they  have  previously  dealt  with  are  ap- 
prehended, even  to  the  extent  of  error,  for  a  rhombus 
is  not  a  square.  That  a  square  is  one  shape  and  a 
rhombus  is  another  shape  is  probably  dimly  under- 
stood by  the  boy;  he  is  giving,  perhaps,  what  he  con- 
ceives to  be  a  genetic  definition  of  a  rhombus,  but  he 
receives  no  marks  for  it  on  our  system  of  marking. 


122  INDUCTIVE   VS.    DEDUCTIVE    METHODS. 

For  his  definition  of  trapezium  he  obtains  three 
marks.  The  rhomboid  he  defines  genetically;  his 
definition  is  worth,  perhaps,  two  marks — one  for 
*  lines'  and  one  for  "two  smallest  lines."  His  defini- 
tion of  diagonal  is  worth  two  marks ;  he  describes  it 
as  a  'line'  and  notes  that  it  goes  from  one  corner  to 
another.  F.  B.  thus  receives  a  total  of  seven  marks, 
which  is  a  little  above  half  the  average  mark  for  the 
class. 

The  two  examples  of  worked  papers,  given  above, 
will  enable  teachers  to  see  the  limits  of  the  mental 
level  with  which  we  are  dealing.  These  boys  are  very 
obviously  much  below  the  first-class  boys  of  the 
higher  grade  school  whose  work  we  considered  in 
the  experiment  previously  described. 

3.     Chronology  of  the  Experiment. 

First  of  all  came  the  inductive  work  with  the 
squares,  triangles,  oblongs  and  diameters  of  circles. 
This  was  done  with  all  the  class.  A  week  or  so  later, 
on  Wednesday,  October  11,  at  10  A.  M.,  following 
immediately  upon  Scripture  lesson,  the  Preliminary 
Test  was  given,  on  the  results  of  which  the  boys  were 
divided  into  two  equal  groups.  Most  of  the  boys  had 
finished  their  work  in  20  minutes,  though  no  one  was 
hurried,  and  one  or  two  took  a  few  minutes  longer. 
On  Thursday,  October  12,  immediately  after  registra- 
tion, the  teacher  of  the  class  taught  one  of  the  groups 
how  to  arrive  inductively  at  the  definitions  of  the 
geometrical  figures  which  they  had  attempted  in  the 
Preliminary  Test.  The  teacher  had  heard  me  teach 
similar  definitions  and  was  well  acquainted  with  the 
method  as  I  employed  it.  The  teaching  took  22  min- 


FIFTH    SERIES   OF   EXPERIMENTS.  123 

utes,  from  2.14  to  2.36  P.  M.  Whilst  the  one  group 
was  being  taught  by  their  own  teacher,  the  other 
group,  under  another  master,  studied  the  definitions 
with  reference  to  the  drawings  of  the  figures  which 
were  appended.  They  knew  that  the  exact  words  of 
the  definitions  were  not  to  be  required,  but  that  they 
might  use  them  if  they  chose.  Both  groups  of  boys 
were  aware  that  they  were  to  be  tested  immediately 
afterwards  on  what  they  had  been  taught  or  learnt. 
The  definitions  given  to  the  *  deductive'  group  ran  as 
follows : 

Definitions  of  Rhombus,  Trapezium,  Rhomboid  and 

Diagonal  of  Square  to  Which  Appropriate 

Drawings  Were  Appended.* 

1.  A  Rhombus t  is  a  figure  with  four  straight 
equal  sides;  the  opposite  sides  are  parallel.    It  has 
four  corners,  two  big  ones  opposite  and  equal,  and 
two  smaller  ones  opposite  and  equal. 

2.  A  Trapezium  is  a  figure  or  shape  with  four 
straight  unequal  sides  and  four  unequal  corners. 

3.  A  Rhomboid  is  a  figure  with  four  straight 
sides.    The  two  long  sides  are  opposite,  equal  and 
parallel.     The  two  short  sides  are  opposite,  equal 
and  parallel.    It  has  four  corners,  two  big  and  two 
small.    The  two  big  ones  are  equal  and  opposite,  and 
the  two  small  ones  are  equal  and  opposite. 


*The  drawings  may  be  seen  on  page  39. 

fPerhaps  a  slight  amendment  might  usefully  have  been  made  in 
this  definition  of  the  rhombus ;  it  is  not  clear  on  this  wording  that 
there  are  two  pairs  of  opposite  sides  which  are  parallel ;  the  form 
of  words  used  in  the  previous  experiment  seems  more  satisfactory. 


124  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

4.  A  Diagonal  of  a  Square  is  a  straight  line 
drawn  from  one  corner  to  the  opposite  corner. 

At  2.40  P.  M.,  a  few  minutes  after  the  teaching  and 
learning,  both  groups  answered  in  writing  the  ques- 
tions: "What  is  a  rhombus  1"  etc.  No  time  restric- 
tions were  laid  down ;  each  boy  was  permitted  to  go 
on  until  he  could  do  no  more,  but  the  superiority  of 
the  pace  of  the  boys  who  had  learnt  the  definitions 
was  evident  in  this  and  in  all  succeeding  exercises. 
After  a  lapse  of  one  week,  at  the  same  hour  in  the 
afternoon,  namely,  2.40,  and  on  the  same  day  of  the 
week,  Thursday,  October  19,  both  groups  answered 
the  questions :  ' '  What  is  a  rhombus  ? ' '  etc.  This  will 
be  referred  to  as  the  first  test  of  deferred  reproduc- 
tion. None  of  the  boys  were  aware  that  they  were 
ever  again  to  be  required  to  answer  these  questions ; 
it  was  only  the  test  of  immediate  reproduction  of 
which  they  had  been  forewarned. 

One  week  later,  again  on  Thursday  at  2.40  P.  M. 
(October  26),  the  boys  worked  a  further  test — a  test 
of  application  to  new  material — and  on  Thursday, 
December  7,  at  2.40  P.  M.,  two  months  after  the  test 
of  immediate  reproduction,  a  second  test  of  deferred 
reproduction  was  given,  in  which  the  old  questions, 
"What  is  a  rhombus?"  etc.,  were  repeated;  and,  as 
before,  the  boys  answered  them  in  writing. 

4.    The  Tests  of  Reproduction. 

These  were  in  all  cases  the  same.  They  consisted, 
as  previously  stated,  of  answers  in  writing  to  the 
questions:  "What  is  a  rhombus?"  etc.  One  or  two 
papers  to  indicate  what  these  boys  could  do  after 
teaching  and  learning  may  be  of  interest. 


FIFTH   SEEIES   OF   EXPEBIMENTS.  125 

Thomas  G ,  aged  13  years  6  months,  one  of  the 

best  boys  who  worked  in  the  deductive  group,  in  his 
exercise  in  immediate  reproduction,  wrote : 

1.  A  Rhombus  is  a  figure  with  four  equal  straight  lines.    The 
opposite  lines  are  parallel.     It  has  four  corners,  one  pair  of  oppo- 
site corners  being  equal  and  the  other  pair  of  opposite  corners  being 
equal. 

2.  A  Trapezium  is  a  figure  with  four  unequal  sides,  and  four 
unequal  corners. 

3.  A  Rhomboid  is  a  figure  with  four  straight  sides,  two  long 
sides!  and  two  short  ones.    The  two  long  ones  are  equal  and  oppo- 
site each  other,  and  the  two  short  ones  are  equal  and  opposite 
each  other.     The  figure  has  four  corners,  two  big  ones  and  two 
little  ones,  The  two  big  ones  are  equal  and  opposite,  and  the  two 
little  ones  are  equal  and  opposite. 

4.  A  diagonal  of  a  square  is  a  straight  line  drawn  from  one 
corner  to  the  opposite  corner. 

This  is  an  excellent  paper ;  the  definition  of  rhom- 
boid, for  example,  where  it  differs  from  the  wording 
of  the  definition  studied,  is  better  than  the  definition 
we  provided.  The  boy  rightly  says  "two  long  and 
two  short"  sides,  before  speaking  of  "The  two  long" 
sides.  Our  own  definition  is  faulty  in  that  respect. 
The  word  'The'  is  not  only  distinguishing,  but  rela- 
tive, and,  indeed,  very  often  distinguishing  because 
relative.  Let  us  mark  the  paper  in  accordance  with 
the  list  of  units  of  correct  description  given  on 
page  —  : 

The  definition  of  rhombus  receives  single  marks 
for  ' figure,'  'four,'  'equal,'  'straight,'  'lines,'  'four/ 
'corners,'  and  eight  marks  for  noting  two  pairs  of 
opposite,  parallel  sides  and  two  pairs  of  opposite 
equal  corners — a  total  of  15  marks.  For  his  defini- 
tion of  trapezium  he  receives  obviously  every  mark 
but  one.  He  has  omitted  'straight'  in  his  descrip- 
tion of  the  sides,  thus  receiving  seven  marks  out  of 
eight.  Every  possible  point  on  our  system  of  mark- 


126  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

ing  is  scored  by  his  definition  of  rhomboid,  with  the 
exception  of  two ;  he  omits  the  two  pairs  of  parallels, 
thus  receiving  18  marks.  The  definition  of  a  diag- 
onal of  a  square  receives  full  marks,  namely,  four. 
F.  G.  's  total  mark  for  his  immediate  reproduction  is 
44,  which  is  much  above  the  average  of  his  group. 

In  his  next  test,  one  week  later,  he  loses  four  marks 
on  his  first  definition,  for  he  now  omits  to  note  the 
two  pairs  of  opposite  parallel  sides.  His  mark  for 
trapezium  remains  unchanged.  In  his  definition  of 
rhomboid  he  now  notes  the  two  pairs  of  parallels, 
which  he  omitted  to  do  in  his  test  of  immediate  re- 
production, and  on  this  occasion  receives  full  marks, 
namely,  20.  The  definition  of  a  diagonal  of  a  square 
remains  unchanged.  F.  G.,  therefore,  has  gone  down 
two  marks  in  one  week.  Let  us  see  how  many  he  has 
lost  seven  weeks  after  this.  The  definition  of  rhom- 
bus suffers  most;  the  parallelism  of  the  opposite 
sides  does  not  reappear,  and  it  is  doubtful  whether 
F.  G.  remembers  that  there  are  two  pairs  of  opposite 
equal  angles,  for  his  expression  is  dubious.  He  has 
now  lost  four  of  the  marks  he  originally  obtained  for 
this  definition.  The  definition  of  trapezium  remains 
unchanged.  In  the  definition  of  rhomboid  two  marks 
are  lost,  for  he  now  omits  to  note  that  there  are  two 
obtuse  angles  and  two  acute  angles.  The  total  mark 
for  this  definition  is  18.  The  definition  of  a  diagonal 
of  a  square  remains  unchanged,  and  receives  four 
marks  as  before.  Two  months  after  learning  the 
definition  F.  G.  receives  40  marks  for  his  reproduc- 
tive test,  against  44  marks  in  his  test  of  immediate 
reproduction,  and  42  marks  in  his  first  test  of  de- 
ferred reproduction,  which  took  place  one  week  after 
he  learnt  the  definitions.  He  loses  very  little ;  he  had 


FIFTH   SERIES   OF   EXPERIMENTS.  127 

evidently  understood  the  definitions  as  well  as  learnt 
them.  Indeed,  his  understanding  is  shown  by  his 
power  of  '  transfer/  for  he  receives  a  very  good 
mark  for  his  application  to  new  material. 

Bearing  in  mind  that  this  pupil  worked  in  the  de- 
ductive group,  let  us  compare  his  work  with  the  cor- 
responding papers  of  one  of  the  best  boys  in  the  in- 
ductive group. 

George  H ,  aged  13  years  9  months,  in  his  test 

of  immediate  reproduction,  wrote : 

1.  A  rhombus  is  a  'figure,'  sides,   four  sides,  all  equal,   four 
angles,  two  opposite  sides  are  parallel,  other  sides  are  parrallel, 
all  sides  are  straight. 

2.  A  trapezium  is  a  figure,   sides,  of  four,   all  unequal,   four 
angles,  angles  unequal,  all  sides  are  straight. 

3.  A  rhomboid  is  a  figure,  of  sides,  four  sides,  two  opposite 
sides  are  equal,  parrallel,  and  has  four  angles,  two  opposite  angles 
are  equal,  sides  straight,  two  opposite  sides  straight,  two  sides  are 
longer  than  the  other  pair  of  sides. 

4.  A  diagonal  of  a  square  is  a  line  from  one  corner  to  the  oppo- 
site corner,  it  is  also  straight. 

Gr.  H.'s  paper  is,  like  F.  G.'s,  an  excellent  one. 
There  is  a  certain  staccato  utterance  which  is  a  little 
irritating,  but  it  is  a  peculiarity  of  his  own  and  is  not 
shared  by  the  members  of  the  inductive  group  gen- 
erally. For  the  definition  of  rhombus  he  receives 
single  marks  for  '  figure,'  'four,7  *  sides,'  '  equal,' 
*  straight,'  'four,'  'angle;'  two  marks  for  noting 
a  pair  of  parallel  sides,  and  that  they  are  opposite 
sides ;  and  one  mark  for  noting  the  other  pair  of  par- 
allel sides ;  but  he  fails  to  note  that  the  other  pair  of 
parallel  sides  are  opposite  also.  He  also  receives 
two  marks  for  noting  that  one  pair  of  angles  are 
equal  and  opposite.  His  total  mark,  therefore,  for 
this  definition  is  12.  His  definition  of  trapezium  re- 
ceives full  marks.  The  definition  of  rhomboid  is  not 


128  INDUCTIVE   VS.   DEDUCTIVE    METHODS. 

so  good.  He  receives  single  marks  for  ' figure,' 
' four,'  'sides,'  'straight,'  'four,'  'angles;'  three 
marks  for  noting  that  one  pair  of  opposite  sides  are 
'equal,'  'opposite'  and  parallel;  two  marks  for  not- 
ing that  one  pair  of  angles  are  equal  and  opposite, 
and  two  marks  for  stating  that  two  sides  are  longer 
than  the  other  two — a  total  of  13  marks.  The  defini- 
tion of  diagonal  scores  full  marks.  G.  H.  thus  re- 
ceives a  total  of  37  marks  for  his  exercise  in  imme- 
diate reproduction. 

In  a  week's  time,  when  he  takes  his  first  test  in 
deferred  reproduction,  he  obtains  one  mark  less.  In 
his  definition  of  rhombus  he  omits  the  parallelism  of 
the  'other  sides,'  losing  a  mark  which  he  had  gained 
the  week  previous.  His  definition  of  trapezium  re- 
mains unchanged.  In  the  definition  of  rhomboid, 
though  it  is  expressed  with  some  slight  differences, 
he  obtains  all  the  marks  which  he  received  before, 
namely,  13.  The  definition  of  diagonal  remains  un- 
changed ;  for  this  he  obtains  four  marks,  as  before, 
making  a  total  of  36  marks. 

Seven  weeks  later  there  is  a  somewhat  more  seri- 
ous loss.  He  still  receives  11  marks  for  the  definition 
of  rhombus,  which  has  remained  unchanged.  His 
definition  of  trapezium  has  improved,  for,  though  it 
contains  no  further  units  of  correct  description  ac- 
cording to  our  scale,  he  notes  that  the  smallest  angle 
is  opposite  the  smallest  side  and  the  biggest  angle  is 
opposite  the  biggest  side.  These  statements  are  not 
quite  clear,  but  indicate  the  commencement  of  a  fresh 
idea  about  the  trapezium.  Two  marks  on  his  pre- 
vious record  are  lost  in  his  definition  of  rhomboid ; 
he  omits  now  that  there  are  two  long  and  two  short 
sides.  The  last  definition  remains  unchanged.  For 


FIFTH   SERIES   OF   EXPERIMENTS.  129 

the  second  paper  in  deferred  reproduction,  there- 
fore, G.  H.  receives  34  marks. 

These  papers  written  by  F.  G.  and  G.  H.,  though 
much  superior  to  the  average  work,  are  typical  in  the 
slowness  with  which  points  like  these  of  definitional 
description  are  forgotten  when  they  have  been  duly 
understood,  and  expressed  in  a  way  which  is  really  a 
result  of  work  on  the  part  of  the  pupil  himself. 

5.    The  Test  of  Application  to  Neiv  Material. 

This,  after  all,  is  the  supreme  test  of  what  teachers 
call  'intelligence. 9  We  have  seen  in  the  two  papers 
given  above  that  the  boy  who  learnt  deductively  knew 
more  of  what  he  had  actually  studied  than  the  boy 
taught  inductively,  not  only  immediately  after  the 
work,  but  after  two  months  had  elapsed ;  and  with  the 
boys  of  this  class  we  shall  find  this  difference  to  be 
true  generally  between  the  boys  of  the  deductive  and 
the  boys  of  the  inductive  groups.  The  older  children 
hitherto — girls  and  boys  graded  as  Standard  VII  and 
upwards — have  not  shown  this  difference,  though  the 
younger  and  less  proficient  children  have.  I  incline 
to  attribute  this  to  the  relative  predominance  of  de- 
ductive work  in  the  usual  teaching  of  this  class, 
whereas  in  the  two  preceding  classes  of  elder  chil- 
dren, both  boys  and  girls,  the  teaching  was  predomi- 
nantly inductive.  Are  we  about  to  find  that  these 
boys  give  us  results  which  differ  from  those  of  the 
older  children  previously  experimented  with,  and, 
indeed,  from  all  the  children  previously  experi- 
mented with,  when  test  is  made  of  their  power  to 
apply  their  knowledge  to  new  material? 

The  test  of  application  was  the  same  as  that  used 


130  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

with  the  Ex- VII  class  in  the  Higher  Grade  Boys' 
School.  Drawings  of  hexagons,  tangents  to  circles, 
pentagons  and  quadrilateral  similar  figures  were 
shown.  The  questions:  "What  is  a  hexagon?" 
"What  is  a  tangent  to  a  circle?"  "What  is  a  penta- 
gon?" and  "In  how  many  ways  does  ABCD  resem- 
ble EFGH?"  were  written  on  the  blackboard  and  the 
children  answered  them  in  writing.* 

I  will  illustrate  what  the  boys  did  by  means  of  two 
papers,  both  above  the  average,  one  from  the  '  de- 
ductive' and  one  from  the  ' inductive'  group. 

Harry  W.,  aged  13  years  6  months,  who  worked  in 
the  *  deductive '  group,  wrote : 

1.  A  hexagon  is  a  figure  with  six  straight  sides  all  of  which  are 
equal,  it  has  also  six  equal  corners  or  angles. 

2.  A  tangent  to  a  circle  is  a  straight  line,  drawn  so  that  it 
touches  the  circumference  of  the  circle. 

3.  A  pentagon  is  a  figure  with  five  straight  sides  and  five  angles, 
all  sides  being  equal  and  all  angles  being  equal. 

4.  A.  b.  c.  d.  is  the  same  as  E.  f.  g.  h.    They  vary  by  the  sides, 
and  the  angles,  if  you  look  at  them  closely  and  then  measure  the 
angles  they  will  all  be  different  on  one  and  all  the  same  as  the 
first  on  the  other.    They  look  different  by  the  size. 

With  the  exception  of  the  last  definition,  this  is  an 
easy  paper  to  mark.  The  definition  of  hexagon  re- 
ceives a  total  of  8  marks.  The  definition  of  tangent 
receives  3  marks.  The  definition  of  pentagon  re- 
ceives 8  marks.  In  the  last  answer  about  the  simi- 
larity of  the  quadrilateral  figures,  it  is  clear  that  H. 
W.  wishes  to  express  the  inequality  of  the  angles  in 
both  figures  and  the  equality  of  the  angles,  each  to 
each,  of  one  figure  with  those  of  the  other,  for  which 
he  receives  5  marks.  Thus  H.  W.,  taught  deductively, 


*The  drawings  may  be  seen  on  page  108.    One  of  the  two  similar 
quadrilaterals  wras  lettered  EFGH  on  this  occasion. 


FIFTH    SEKIES    OF   EXPERIMENTS.  131 

scores  24  marks  for  his  test  of  application  to  new 
material. 

Frank  C ,  aged  13  years  2  months,  who  was 

taught  inductively,  wrote : 

1.  A  Hexagon  is  a  six  straight  sided  figure,  having  all  sides 
equal,  it  has  six  angles  equal,  larger  than  right  Angles. 

2.  A  Tangent  to  a  circle  is  a  line  which  is  straight  and  is  just 
touching  the  boundary  of  a  circle. 

3.  A  Pentagon  is  a  five,  equal,  straight  sided  figure,  it  has  five 
equal  angles  larger  than  right  angles. 

4.  Both  have  four  sides. 
Both  have  four  angles. 

Both  have  four  angles  which  are  larger  than  right  angles. 

A  angle  equals  E  angle. 

B  "        F  angle. 

C      "  G  angle. 

D      "          "        II  angle. 

Both  have  sides  with  the  same  slope. 

Both  are  placed  on  the  same  side. 

The  definition  of  hexagon  receives  a  total  of  nine 
marks;  the  definition  of  tangent  three  marks;  and 
that  of  pentagon  nine  marks.  The  last  answer  is  more 
difficult  to  mark.  Both  figures  have  '  sides ; '  this  car- 
ries one  mark.  There  are  four  sides  in  both  figures ; 
this  carries  another  mark.  Similarly,  "Both  have 
four  angles ' '  carries  two  marks.  The  next  statement 
is  wrong;  it  is  not  true  that  both  have  four  angles 
which  are  larger  than  right  angles.  Then  there  are 
four  marks  for  noting  the  equality  of  the  angles,  each 
to  each,  and  four  marks  for  noting  that  the  sides  of 
the  figures  are  parallel.  One  further  mark  is  gained 
by  F.  C.  's  statement  that  both  the  figures  are  on  the 
same  side  (of  the  base).  This  answer,  therefore,  re- 
ceives a  total  of  13  marks.  The  paper  is  an  excellent 
one,  and  carries  a  total  of  34  marks ;  it  is,  in  fact,  one 
of  the  best  papers  worked  in  either  group  in  the  test 
of  application  to  new  material. 


132  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

Lest  the  reader  should  carry  away  a  quite  exag- 
gerated notion  of  the  power  of  application  of  these 
pupils  (I  am  using  the  expression  ' application'  in 
the  strictest  sense),  I  propose  to  give  one  further 
paper  by  a  boy  who  worked  in  the  deductive  group 
and  made  very  little  application  of  his  knowledge. 
He  obtained  37  marks  in  his  test  of  immediate  repro- 
duction and  34  marks  a  week  after.  But  he  obtained 
a  very  poor  mark  when  he  worked  on  new  material, 
and  seven  weeks  later  he  sank  to  23  marks  when 
tested  on  his  old  knowledge.  There  are  evidently 
some  boys  who  learn  quickly  and  forget  quickly.  The 
pedagogical  error,  now  happily  being  rectified  by 
psychologists,  has  been  to  regard  these  boys  as  the 
rule  rather  than  the  exception.  This  boy,  George  L., 
aged  12  years  4  months,  wrote : 

1.  A  Hexagon  is  a  six  sided  figure.    Each  of  the  six  sides  are 
straight  equal  and  opposite  and  Paralled. 

2.  A  tangent  of  a  circle  is  a  straight  line  drawn  which  is  slant- 
ing and  the  circle  stands  on  it. 

3.  A  Pentagon  is  a  figure  with  five  sides,  they  are  all  straight. 
The  Three  small  ones  are  equal  and  opposite,  and  the  two  long 
ones  are  equal  and  opposite. 

4.  a.  b.  c.  d's.  has  two  straight  long  sides  equal  and  the  other 
two  sides  unequal  E.  f.  g.  h's  has  two  long  straight  sides  equal, 
and  the  other  two  unequal  only  a.  b.  c.  d  is  smaller  than  E.  f.  g.  II. 

" Equal  and  opposite"  has,  unfortunately,  trans- 
ferred itself  too  successfully.  For  his  definition  of 
hexagon  he  receives  5  marks.  "Each  of  the  sides  are 
opposite  and  parallel"  is  considered  to  be  too  con- 
fused to  gain  positive  marks,  but  is  not  regarded  as 
involving  'bad  errors.'  The  definition  of  tangent 
gains  2  marks  only ;  the  latter  part  of  his  definition 
was  drawn  from  one  figure  only.  The  statements 
that  the  tangent  is  slanting  and  that  the  circle  stands 


FIFTH   SERIES   OF   EXPERIMENTS.  133 

on  it  were  not  true  of  all  the  tangents  drawn,  and  are 
considered  'bad  errors.'  In  his  definition  of  penta- 
gon he  receives  4  marks  only.  It  is  considered  a  'bad 
error'  to  say  that  "three  small  ones  are  opposite." 
No  positive  marks  are  allowed  for  saying  that  "three 
are  equal"  and  "two  are  equal,"  and  it  is  counted  an 
error  to  say  there  are  "two  long"  and  "three  small" 
sides.  In  his  last  answer  Gr.  L.  receives  2  marks; 
both  the  figures  have  '  sides, '  and  in  each  case  two  are 
longer  than  the  remaining  two.  But  none  of  them 
were  equal;  though,  as  two  of  them  were  not  very 
different  in  length,  the  statement  was  not  accounted 
a  'bad  error.'  The  statement  as  to  the  size  of  the 
two  figures  is  irrelevant;  the  boys  were  asked  for 
'resemblances,'  not  for  differences.  This  is  one  of 
the  worst  papers  in  the  class.  The  boy  had  acquired 
the  knowledge  of  the  definitions  of  rhombus,  etc.,  but 
he  could  not  apply  it,  and  he  speedily  forgot  it. 

Possibly,  with  these  examples  before  him,  the 
reader  may  find  greater  interest  in  the  tabulated 
results,  which  I  now  give. 

6.    Results, 
(a)     Of  the  Preliminary  Tests. 

In  the  Preliminary  Test  the  highest  mark  gained 
by  any  boy  was  19,  the  lowest  was  6,  and  the  average 
mark  was  12.25.  There  were  16  boys  in  the  group 
deductively  taught  and  16  boys  in  the  group  induct- 
ively taught.  The  average  mark  of  the  boys  of  the 
first  group  was  12.25  (mean  variation  3.0),  and  of 
those  in  the  second  group  was  12.25  (mean  variation 
3.0).  The  correlation  between  the  total  results  of  the 
corresponding  boys  in  the  two  groups  was  practi- 


134  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

cally  perfect,  amounting  to  +  .97  on  the  product- 
moment  formula.  In  so  far  as  one  test  can  in  any 
way  be  satisfactory  as  a  basis  of  the  division  of  a 
class  into  equal  groups,  it  seems  fair  to  suppose  that 
an  adequate  division  has  been  made.  These  boys,  it 
will  be  remembered,  had  had  some  special  inductive 
teaching  concerning  the  square,  triangle,  etc.,  though 
I  should  not  describe  the  general  methods  of  their 
teacher  as  predominantly  inductive.  I  incline  to 
think  this  special  inductive  propaedeutic  may  have 
been  an  advantage  to  us  in  making  the  division,  but 
it  may,  I  fear,  serve  to  throw  some  bias  on  the  in- 
ductive side  and  unduly  favor  the  inductive  group. 
We  may,  however,  remember  that  we  have  three  ex- 
periments already  described  in  which  no  such  propae- 
deutic was  given. 

(b)     Of  Immediate  Reproduction. 

What  marks  did  the  two  groups  obtain  immedi- 
ately after  the  teaching  and  learning?  In  two  pre- 
vious experiments  with  older  children,  girls  as  well 
as  boys,  the  group  taught  inductively  appeared  to 
advantage  from  the  first.  Is  that  also  the  case  with 
these  Standard  VII  boys!  We  can  say  quite  defin- 
itely that  it  is  not.  The  average  mark  obtained  by 
the  boys  of  the  deductive  group  was  34.2  (mean  vari- 
ation 6.0),  and  of  the  inductive  group  31.4  (mean 
variation  4.5). 

This  difference  between  the  means  and  its  prob- 
able error  justify  us  statistically  in  asserting  the 


FIFTH   SEEIES   OF   EXPERIMENTS. 


135 


-Deductive  Group.  N 

/  Inductive  Group.  v 

Av.  mark 

Av.  mark 

Average 

in  imme- 

Average 

in  imme- 

mark in 

diate 

mark  in 

diate 

>.       prelimi- 

repro- 

No. 

prelimi- 

repro- 

nary 

duction 

of 

nary 

duction 

s.        test. 

test. 

boys. 

test. 

test. 

17.0 

38.0 

4 

17.0 

34.5 

12.4 

32.7 

7 

12.5 

29.6 

8.2 

33.2 

5 

8.0 

31.4 

existence  of  a  general  tendency  in  favor  of  the  l  de- 
ductive' group.  The  superiority  of  the  work  of  the 
deductive  group  in  immediate  reproduction  may 
also  be  shown  compendiously  in  the  following  table : 

Table  XXVI,  showing  the  work  of  the  Deductive  and  Inductive 
Groups  compared,  section  ~by  section,  in  the  Preliminary  Test 
and  the  Test  of  Immediate  Reproduction. 


Marks  in        No. 
preliminary       of 
test.  boy 

Over  15 4 

10  to  15 7 

5  to  10 5 


(c)     Correspondence  Betiveen  Immediate  and  De- 
ferred Reproduction. 

But,  after  all,  the  important  question  in  education 
is  not  so  much  what  can  be  done  by  pupils  immedi- 
ately after  they  have  just  been  taught,  but  what  they 
can  do  some  time  afterwards.  Do  they  remember 
what  they  once  knew,  and  how  far  can  they  apply 
their  knowledge?  To  the  second  of  these  questions 
I  hope  to  give  an  answer  when  dealing  with  the  re- 
sults of  the  test  on  new  material.  Let  me  turn  for  a 
while  to  the  first,  and  let  me  break  it  up  into  a  number 
of  constituent  questions.  The  boys  gain  certain 
marks  immediately  after  teaching  and  learning. 
What  do  they  gain  a  week  later,  and,  more  important 
still,  what  do  they  gain  two  months  later  ? 


136  INDUCTIVE   VS.   DEDUCTIVE    METHODS. 

In  a  rough  way  we  can  find  the  answers  to  our 
questions  in  the  following  table : 

Table  XXVII,  shoiving  the  work  of  the  Inductive  and  Deductive 
Groups  compared,  section  by  section,  in  the  Tests  of  Immediate 
and  Deferred  Reproduction. 

Deductive  Group. 

/ Average  Marks. * 

Imme-  Deferred  Deferred 

diate  repro-  repro 

Marks  for                         No.             repro-  duction,  duction, 

immediate                          of            duction  first  second 

reproduction.                     boys.            test.  test.  test. 

40  and  over 5                41.4  38.6  38.4 

35  to  40 3                37.7  38.0  32.7 

30  to  35 4                30.8  28.3  25.8 

25  to  30 3                26.7  22.3  23.0 

Below  25 1                22.0  20.0  19.0 

Inductive  Group. 

, Average  Marks. \ 

Imme-  Deferred  Deferred 

diate  repro-  repro- 

Marksfor                         No.             repro-  duction,  duction, 

immediate                          of            duction  first  second 

reproduction.                     boys.            test.  test.  test. 

40  and  over 0                —  —  — 

35  to  40 6                37.0  33.7  31.7 

30  to  35 3                32.0  29.3  27.0 

25  to  30 6                27.3  22.8  24.5 

Below  25 1                20.0  17.0  24.0 

The  conclusions  seem  clear.  The  Inductive  Group 
contains  no  boys  at  all  equal  to  the  highest  section 
of  the  Deductive  Group.  The  best  boys  in  the  In- 
ductive Group  correspond  to  the  second  section  of 
the  Deductive  Group,  but  even  then  they  are  inferior 
to  that  section,  both  in  the  immediate  and  deferred 
tests.  The  work  done  in  immediate  reproduction 
may  be  very  well  taken  as  representative  of  what  the 
work  will  be  later  on  in  exercises  of  this  kind,  for  the 
various  sections  into  which  the  groups  are  divided 


FIFTH    SERIES   OF   EXPERIMENTS.  137 

retain  their  relative  positions  throughout  the  whole 
experiment.  Calculated  exactly,  the  correlation  co- 
efficients between  the  results  of  Immediate  Repro- 
duction and  those  of  the  first  Deferred  Reproduction 
Test  in  the  Deductive  Group  is  +  .804,  and  between 
Immediate  Reproduction  and  the  second  Deferred 
Reproduction  Test  (two  months  later)  is  +  .859. 
The  corresponding  figures  for  the  Inductive  Group 
are  +  .616  and  +  .619. 

Summarizing  the  results  and  treating  the  groups 
as  wholes,  the  averages  and  variabilities  are  as  fol- 
low: 

Table  XXVIII,  shoiring  the  ivorTc  of  the  Inductive  and  Deductive 

Groups  compared  in   the  Tests  of  Immediate  and   Deferred 
Reproduction. 

Imme-  First  Second 

diate  deferred  deferred 

repro-  repro-  repro- 

Deductive  Group:          duction.  duction.  duction. 

Average  mark 34.2  32.5  30.1 

M.  V 6.0  6.3  7.0 

Inductive  Group : 

Average  mark 31.4  27.8  27.6 

M.  V 4.5  5.1  4.1 

The  Deductive  Group  has  outdistanced  the  In- 
ductive Group  quite  clearly,  both  in  immediate  and 
deferred  reproduction,  not  only  in  positive  marks, 
for,  perhaps,  I  ought  to  add,  it  has  also  made  fewer 
'bad  errors.'  It  is  the  third  result  in  which  this  has 
been  found  to  be  the  case.  We  shall,  therefore,  again 
have  to  admit  the  contention  urged  against  induct- 
ive methods  in  the  earlier  chapters  of  this  mono- 
graph. We  must  certainly  conclude  that,  in  exami- 
nations on  precisely  what  has  been  taught  or  learnt, 
children  taught  by  what  we  have  called  deductive 
methods  may  be  more  successful  than  children  taught 


138  INDUCTIVE   VS.   DEDUCTIVE   METHODS. 

inductively.  Also  we  see  that  children  need  not  be 
young  to  be  taught  successfully  by  deductive  meth- 
ods. Let  us  now  turn,  however,  to  the  Test  of  Ap- 
plication to  New  Material  and  see  whether  the  same 
relation  between  the  two  groups  holds  there. 

(d)     Results  of  the  Test  on  New  Material. 

We  have  seen  that  for  purposes  of  immediate,  and 
even  of  deferred,  reproduction  the  more  mechanical 
method  has  shown  itself  superior  to  the  less  mechan- 
ical. Is  the  same  relationship  retained  between  the 
two  groups  when  the  test  is  no  longer  one  of  simple 
reproduction,  but  requires  a  transfer  of  knowledge 
or  method  to  analogous  material?  We  can  say  at 
once  that  the  same  relation  is  not  maintained.  The 
inductive  group  now  comes  to  the  front,  but  the  dif- 
ference between  the  means  of  the  two  groups  is  a 
small  one  and  the  variability  of  the  averages  is  high. 
The  deductive  group  scores  an  average  mark  of  20.5 
(mean  variation  5.9),  and  the  inductive  group  an 
average  mark  of  21.1  (mean  variation  4.4).  But  let 
us  look  a  little  more  closely  into  the  composition  of 
these  averages : 

Table  XXIX,  shcnving  the  worlc  of  the  Inductive  and  Deductive 
Groups  compared  in  Immediate  Reproduction  and  in  the  Test 
on  New  Material. 

, — Deductive  Group. — ^    , Inductive  Group. \ 

Marks                          Average  Marks  for  Average  Marks  for 
in  imme-                           Imme-  Imme- 
diate                 No.        diate  No.  diate 
repro-                  of        repro-        New  of  repro-         New 
duction.               boys,    duction.  material,  boys,  duction.    material. 

Over  35 8           40.0           23.0  6  37.0           24.8 

30  to  35 4           30.8           17.8  3  32.0           19.3 

25  to  30 3           26.7           19.7  6  27.3           17.5 

Below  25 1           22.0           12.0  1  20.0           15.0 


FIFTH   SEEIES   OF   EXPEBIMENTS.  139 

The  figures  certainly  suggest  a  superiority  on  the 
side  of  the  inductive  group  in  three  of  the  corre- 
sponding sections  into  which  the  groups  are  divided ; 
and  the  regular  decline  of  the  figures  in  both  groups 
(with  the  exception  of  the  average  of  19.7  in  the  third 
section  of  the  Deductive  Group)  would  appear  to  in- 
dicate that  there  is  a  general  tendency  in  favor  of 
correlated  transfer  in  the  Inductive  rather  than  in 
the  Deductive  Group.  The  coefficient  of  correlation 
between  the  results  of  the  Inductive  and  Deductive 
Groups,  when  tested  on  new  material,  is,  however, 
not  very  high.  With  high  variability  as  well,  this 
involves  a  high  probable  error.  So  that  we  may  con- 
clude in  this  case  merely  that  the  Inductive  Group 
does  better  work  on  the  whole  than  the  Deductive 
Group,  but  we  have  not  the  usual  statistical  justifi- 
cation that  there  is  a  strong  general  tendency  in  that 
direction.  We  shall,  however,  hardly  feel  disposed 
to  attribute  the  superiority  of  the  Inductive  Group 
to  chance,  since  in  every  one  of  the  five  experiments, 
with  different  teachers,  with  children  of  different 
ages,  of  different  abilities  and  of  different  sexes,  we 
have  found  the  inductively  taught  group  the  more 
competent  when  tested  on  the  power  of  application 
to  new  material. 


IX.    GENERAL   SUMMARY. 

In  five  different  schools  in  different  parts  of  Lon- 
don, attended  by  children  varying  in  social  class,  ex- 
periments have  been  made  to  test  the  relative  values 
of  *  inductive'  and  '  deductive'  methods  of  teaching 
as  applied  to  geometrical  definition.  Both  girls  and 
boys,  of  ages  ranging  from  8  to  15  years,  were  set  to 
do  the  work.  The  main  problems  were  two  in  num- 
ber. In  the  first  place,  an  attempt  was  made  to  dis- 
cover which  of  the  two  methods  gave  the  better  re- 
sults when  the  children  were  tested  on  precisely  what 
they  had  been  taught  or  had  learnt.  In  the  second 
place,  an  endeavor  was  made  to  find  out  which  of  the 
two  methods  gave  the  better  results  when  the  chil- 
dren were  tested  on  new  material. 

The  answer  to  the  first  of  these  two  questions  was 
not  the  same  in  all  of  the  five  schools  tested.  In  three 
of  them,  two  of  the  three  boys '  schools  and  one  of  the 
two  girls'  schools,  the  conclusion  was  unambiguously 
in  favor  of  the  '  deductive  and  memoriter'  method. 
This  was  the  case  with  the  younger  and  less  profi- 
cient boys  and  girls,  and  at  first  sight  it  looked  as  if 
age  were  an  important  factor  in  the  production  of 
this  result,  but  the  same  result  was  obtained  with  a 
class  of  boys  who  were  much  older,  so  that  age  was 
certainly  not  the  only  factor  of  differentiation.  In 
two  classes,  the  oldest  class  of  boys  and  the  oldest 
class  of  girls  who  did  the  work,  the  inductive  method 

140 


GENERAL   SUMMARY.  141 

was  just  as  successful  as  the  ' deductive,'  even  for 
purposes  of  exact  reproduction,  immediately  after- 
wards, of  what  had  been  taught  or  learnt.  There 
were  some  indications  that  the  children  inductively 
taught  lost  rather  less  of  what  they  had  known  than 
those  deductively  taught  when  they  were  tested  some 
time  afterwards;  but,  on  the  whole,  the  tests  of  de- 
ferred reproduction  gave  the  same  comparative  re- 
sults as  those  of  immediate  reproduction.  The  im- 
portance of  this  consideration  in  testing  school 
methods  where  exact  reproduction  is  required  is 
obvious.. 

The  answer  to  the  second  of  the  two  main  issues 
was  the  same  in  all  of  the  five  schools  tested.  The 
children  who  were  taught  ' inductively'  did  better 
work  than  those  taught  ' deductively'  in  every  case 
when  they  were  required  to  apply  themselves  to  new 
material. 

This  research,  therefore,  offers  an  experimental 
justification  of  what  are  known,  among  teachers,  as 
'  intelligent '  methods  of  teaching,  and  of  the  superior 
' transfer'  effect  of  certain  methods. 

Many  pedagogical  corollaries  may  be  drawn  from 
the  experiments,  but  it  will  be  sufficient  in  this  place 
to  emphasize  a  consideration  already  alluded  to  in 
the  body  of  the  text. 

Examinations,  whether  internal,  that  is,  conducted 
from  within  by  the  school  authorities,  or  external, 
that  is,  conducted  by  external  educational  authori- 
ties, should  always  include  questions  on  subject-mat- 
ter which  is  not  identical  with  that  set  down  in  the 
syllabuses  of  instruction  if  the  examination  is  to  test 
good  method  in  teaching.  But  if  the  tests  are  to 
serve  any  useful  pedagogical  purpose,  the  new  mate- 


142  INDUCTIVE   VS.    DEDUCTIVE    METHODS. 

rial,  though  it  should  not  be  identical,  ought  to  be 
analogous  to  that  which  has  been  dealt  with  in  the 
school  curriculum.  Questions  on  new  analogous  ma- 
terial are  probably  the  best  questions  of  all  (if  the 
same  set  of  questions  be  required  to  serve  a  double 
purpose),  for  they  test,  with  fair  adequacy,  whether 
the  work  set  down  in  the  syllabuses  has  been  effi- 
ciently done,  and  they  also  test,  with  admirable  ade- 
quacy, whether  the  methods  by  which  the  school  work 
has  been  done  were  such  as  to  give  the  pupil  power 
to  apply  his  knowledge. 


INDEX 

'Bad'  errors,  meaning  of,  36. 

and  mechanical  method,  51. 
Chance  or  Variability,  7. 

Children's  Definitions,  spontaneous,  27,  28,  57,  71,  72,  73, 
74,  102,  103,  120,  121. 

after  teaching  and  learning,  76,  77,  78,  80,  81,  82,  125, 
127. 

of  new  analagous  material,  40,  83,  84,  85,  86,  88,  90, 

109,  111,  130,  131,  132. 
Circle,  definition  of  diameter  of,  29,  32. 

drawing  of  diameter  of,  24. 

Classes  taking  the  experiment,  20,  23,  55,  69,  100,  119. 
Co-operation  of  Teachers,  4. 
Correlation  coefficients,  9, 10,  50,  62,  66,  91,  96,  115,  137. 

value  of,  30. 
Deductive  Method,  method  of  learning  by,  37. 

method  of  testing,  19. 

shown  to  be  the  better,  46,  93,  134,  136,  137. 
Deferred  Reproduction,  44,  47,  63,  65,  92,  107,  124,  135. 
Definitions,  'real/  26,  27. 

arguments  in  favor  of  deductive  treatment  of,  17. 

arguments  in  favor  of  inductive  treatment  of,  18. 

units  of  marking  of,  28,  29,  41,  42,  43. 

as  learnt  deductively,  31. 

as  learnt  inductively,  32. 

children's  spontaneous,  after  teaching  and  learning, 
of  new  analagous  material,  see  Children's  Defini- 
tions. 

of  diameter  of  circle,  29,  32. 

of  hexagon,  113. 

of  oblong,  29,  32. 

143 


144  INDUCTIVE  VS.  DEDUCTIVE  METHODS. 

of  pentagon,  113. 

of  rhomboid,  42, 106,  123. 

of  rhombus,  41,  105,  123. 

of  square,  25,  28,  32,  35. 

of  diagonal  of  square,  39,  106,  124. 

of  tangent  to  circle,  113. 

of  trapezium,  42,  106,  123. 

of  triangle,  28,  32. 

Demonstrative  Geometry,  introduction  to,  17,  27. 
Diagonals  of  Squares,  39,  106,  124. 
Diameter  of  Circle,  definition  of,  29,  32. 

drawing  of,  24. 

Durability  of  knowledge,  19,  37,  48,  49,  64,  66,  95. 
Education,  method  of  settling  disputed  questions  in,  16. 
Educational  Science,  3. 
Equal  groups,  how  formed,  30,  45,  56,  61,  92,  133. 

use  of,  52. 
Errors,  'bad/  meaning  of,  36. 

and  mechanical  method,  51. 
Errors,  method  of  correcting  inductively,  34. 

'probable  errors/  method  of  determining,  8-10. 

in  spelling  not  counted,  43. 
Experiment,  use  of,  16. 
Experimental  Pedagogy,  1. 
Geometrical  Definitions,  see  Definitions. 
Geometrical  Teaching,  alleged  cause  of  'chaos  in/  18. 
Geometry,  Demonstrative,  introduction  to,  17,  27. 
Groups,  equal,  how  formed,  30,  45,  56,  61,  92,  133. 

use  of,  52. 
Hexagon,  definition  of,  113. 

drawings  of,  108. 
Immediate  Reproduction,  19,  44,  47,  63,  65,  92,  107,  124, 

135. 
Inductive  Method,  an  objection  to,  14. 


INDUCTIVE  VS.  DEDUCTIVE  METHODS.  145 

arguments  for,  18. 

method  of  learning  by,  32. 

method  of  correcting  by,  34. 

method  of  testing,  19. 

shown  to  be  the  better,  51,  52,  64,  66,  67,  98,  116,  117, 

138. 
Intelligence,  meaning  of,  18. 

test  of,  38,  129. 

training  of,  53. 
Knowledge,  durability  of,  19. 

relation  between  quickness  and  permanence,  37,  48, 

49,  64,  66,  95. 

Marks,  positive  and  negative,  36,  38. 
Material,  'new,'  see  'New  Material.' 
Negative  marks,  36,  38. 
'New  material/  meaning  of,  53,  54. 

application  to,  19,  38,  50,  67,  96,  107,  116,  129,  138. 
'New  methods/  general  tendency  of,  13,  14. 

alleged  disadvantages  of,  14. 
Novelty,  influence  of,  70. 
Oblong,  definition  of,  29,  32. 

drawings  of,  24. 
Pedagogy,  Experimental,  1. 
Pentagon,  definition  of,  113. 

drawings  of,  108. 
Positive  Marks,  36,  38. 
Practice  versus  Theory,  11,  13,  15. 
'Probable  Errors/  method  of  determining,  8-10. 
Reproduction,  deferred,  44,  47,  63,  65,  92,  107,  124,  135. 

immediate,  19,  44,  47,  63,  65,  92,  107,  124,  135. 
Rhomboid,  definition  of,  42, 106, 123. 

drawings  of,  39. 
Rhombus,  definition  of,  41, 105, 123. 

drawings  of,  39. 


146  INDUCTIVE  VS.  DEDUCTIVE  METHODS. 

School  Classes  taking  the  experiment,  20,  23,  55,  69,  100, 

119. 

Science,  Educational,  3. 
'Science'  of  Education,  4. 
Spelling  errors  not  counted,  43. 
Spontaneous  definitions,  27,  28,  57,  71,  72,  73,  74,  102,  103, 

120,  121. 
Square,  definition  of,  25,  28,  32,  35. 

diagonals  of,  39, 106, 124. 

drawings  of,  24. 
Tangent  to  Circle,  definition  of,  113. 

drawings  of,  108. 
Teaching,  the  divergence  of  Theory  and  Practice,  11. 

breach  between  Theory  and  Practice,  13,  15. 

unintelligent,  reaction  against,  70. 
Teachers,  co-operation  of,  4. 
Theory  versus  Practice,  11,  13,  15. 
Time  taken  for  the  exercises,  44,  59,  75,  104,  106,  122. 
Trapezium,  definition  of,  42,  106,  123. 

drawings  of,  139. 

Unintelligent  teaching,  reaction  against,  70. 
Unsophisticated  material,  20. 
Variability  or  Chance,  7. 


Cfrucaltoutl 


hg  <$itg 


WARWICK  &  YORK,  Inc. 

,  1.  **  A. 


Moto- 
Sensory 
Develop- 
ment 

Observations 
on    the    First 
Three  Years 
of  a  Child. 


By 

GEORGE 
V.  N. 

DEARBORN 


Price: 

I2mo, 

215  +  vi  pages , 

frontispiece. 

$1.50. 


Few  subjects  are  of  greater  interest  to 
the  parents  of  young  children  or  to 
school  teachers  with  the  truly  scientific 
spirit  of  their  profession  than  the  evolu- 
tion of  a  child's  mechanism  of  efficiency. 
To  the  psychologist,  and  to  a  less  extent 
to  the  physiologist,  acquaintance  with  the 
average  course  of  this  human  unrolling 
is  clearly  a  technical  necesstiy.  All  these 
surely  should  welcome  every  competent 
new  account  of  the  first  three  years  of 
human  life. 

This  book,  as  its  name  implies,  dis- 
cusses both  the  motor  and  the  sensory 
development  of  an  average  child.  It  con- 
sists of  careful  observations  of  the  steps 
in  individual  evolution  with  the  addition 
of  numerous  notes  and  brief  theoretic 
discussions  of  the  observations.  The 
chief  emphasis  has  been  put  on  the  be- 
ginnings of  voluntary  movement  and  on 
the  forerunning  phenomena.  These  are 
considered  from  both  the  physiologic  and 
psychologic  points  of  view. 

The  affective  side  of  child-development 
is  more  fully  treated  than  are  the  purely 
intellectual  processes,  although  the  moto- 
sensory  evolution  of  ideation  as  exhibited 
in  learning  to  talk  is  as  amply  considered 
as  circumstances  allowed  and  as  was  ex- 
pedient. 

A  feature  of  the  book  Is  a  careful 
chronologic  epitome  of  the  observed  de- 
velopment, perhaps  more  detailed  than  in 
any  work  since  the  pioneer  treatise  of 
Preyer.  This  is  given  in  two  tables  of 
considerable  length,  one  of  them  ar- 
ranged alphabetically  and  the  other  by 
weeks.  For  purposes  of  reference  these 
tables  will  be  found  of  value. 

Throughout  the  book  there  is  continual 
reference  to  the  temporal  and  other  re- 
lationships of  mental  development  as 
noted  in  similar  accounts  by  Preyer,  Dar- 
win, Shinn,  Moore,  Major  and  others. 
These  notes  facilitate  the  use  of  the  book 
for  pedagogical  purposes,  and  they  also 
enable  parents  to  judge  more  accurately 
of  the  natures  of  their  children  in  com- 
parison with  the  average. 


WARWICK  &  YORK,  Inc.,  BALTIMORE,  1CD. 


Spelling 

Efficiency 

in 

Relation 

to  Age, 

Grade 

and  Sex, 

and  the 

Question 

of 

Transfer 


An  Experi- 
mental and 
Critical 
Study   of  the 
Function  of 
Method  in  the 
Teaching  of 
Spelling. 


By 

J.   E. 

WALLACE 
WALLIN 


Price: 

12wo,   cloth, 
vi,  91  pages. 

n.26. 


There  are  few  elementary  school  tub- 
jects  in  which  inefficiency  is  more  surely 
detected  and  reprobated  in  later  life,  and 
in  the  teaching  of  which  the  elementary 
schools  are  charged  with  more  extrava- 
gant waste  of  time,  than  spelling.  7.22 
per  cent,  of  the  time  of  the  child  in  the 
elementary  schools  in  ten  of  our  largest 
cities  is  devoted  to  the  study  of  spelling, 
and  yet  the  complaint  continues  to  be 
almost  universally  voiced  that  the  ele- 
mentary and  secondary  school  graduates 
have  not  learned  how  to  spell. 

School  superintendents  and  teachers 
have  felt  the  justice  and  sting  of  these 
criticisms,  and  have  attempted  to  pro- 
vide a  remedy  either  by  increasing  the 
time  devoted  to  spelling  or  by  changing 
the  methods  of  teaching.  The  results, 
however,  have  not  in  all  cases  proved 
satisfactory. 

Dr.  Wallin,  who  has  been  offering 
courses  in  educational  psychology  and  the 
principles  of  teaching  in  schools  of  edu- 
cation for  a  number  of  years,  points  out 
briefly  in  this  monograph  some  of  the 
fallacies  involved  in  the  exclusive  use  of 
the  incidental  method  of  teaching  spell- 
ing, based  upon  the  psychological  prin- 
ciples which  condition  the  reduction  of 
mechanical  subject-matter  to  the  plane  of 
automatism  (spelling  is  of  an  instru- 
mental nature).  By  means  of  the  re- 
sults of  the  very  researches  made  in  the 
past  to  demonstrate  the  adequacy  of  the 
incidental  method,  it  is  shown  that  its 
use  has  not  justified  the  claims  made  in 
its  behalf.  On  the  other  hand,  the  su- 
periority of  a  spelling  drill  technique, 
based  upon  the  laws  of  habit  formation, 
is  shown,  partly  by  the  author's  own  in- 
vestigation and  partly  by  the  results  of 
a  thoroughgoing  application  of  the  meth- 
od under  control  conditions  during  four 
years  in  a  large  school  system. 

The  book  also  discusses  the  relation  of 
spelling  efficiency  to  age,  grade  and  sex ; 
the  facts  derived  from  the  tests  are  sup- 
ported by  numerous  tables,  a  number  of 
practical  conclusions  are  offered,  and  a 
bibliography  is  appended. 


When 
Should  a 
Child 
Begin 
School? 

An   Inquiry 
Into   the 
Relation 
Between    the 
Age  of  Entry 
and   School 
Progress. 

By 

W.  H. 
WINCH 


Price: 


108  pages. 
$1.25. 


Few  educational  questions  have  excited 
more  general  interest  in  recent  years 
than  that  of  the  age  at  which  children 
siiould  commence  their  attendance  at 
school.  On  the  one  side  we  have  the 
rule-of-three  conclusion,  felt  rather  than 
expressed  as  an  inference,  that  the  more 
teaching  the  child  gets  and  the  sooner 
he  begins  school  the  more  progress  he  is 
sure  to  make.  On  the  other  we  have  had 
a  strong  feeling,  now  growing  in  inten- 
sity and  range,  that  attendance  in  school, 
particularly  in  England,  begins  too  early 
and  that  there  is  an  educational  disad- 
vantage in  commencing  as  soon  as  the 
children  of  Great  Britain  do.  While  this 
investigation  by  Mr.  Winch  has  special 
reference  to  England,  where  the  school 
hfe  begins  at  a  much  earlier  period  than 
in  either  America  or  Germany,  the  re- 
sults set  forth  by  the  author  are  of  vital 
interest  to  all  who  have  to  do  with  the 
education  of  children. 

The  effect  of  age  of  entry  is  considered 
from  several  points  of  view :  1.  Does 
early  entry  at  school  enable  the  pupil  to 
make  more  rapid  advancement  in  school 
standing  than  entry  at  a  later  age?  In 
other  words,  in  a  given  grade  are  those 
pupils  who  entered  school  earlier  found 
to  constitute  the  younger  portion  of  the 
class?  2.  In  the  same  grade  some  pupils 
may  be  doing  work  of  a  high  degree  of 
efficiency,  others  work  of  an  inferior 
qualitv.  To  what  extent  does  early  entry 
correlate  with  high  efficiency  when  tested 
by  examinations?  3.  How  far  does  early 
entry  depend  upon  social  circumstances? 
4.  What  is  the  influence  of  early  entry 
upon  the  subsequent  behavior  of  pupils 
and  upon  their  attentiveness  to  school 
work? 

The  results  of  Mr.  Winch's  inquiry  are 
now  published  for  the  first  time.  Some 
of  them  have  been  privately  circulated, 
and  a  few  of  the  tables,  together  with 
the  methods  employed,  were  discussed 
some  years  ago  at  a  meeting  of  the  In- 
spectors of  the  Education  Committee  for 
London. 


WARWICK  &  YORK,  Inc.,  BALTIMORE,  UP. 


Mental 
Fatigue 

"Die 

Geistige 

Ermtidung.' 


Bj 

MAX 
OFF1TER, 

Translated 
from   the 
German   by 
GUY 

MONTBOSE 
WHIPPLS 


Price: 

12mof   cloth, 

viii,  133  page*. 

$1.25. 


This  noteworthy  monograph  is  a  com- 
prehensive exposition  of  the  nature  of 
mental  fatigue,  of  the  methods  proposed 
for  measuring  it,  and  of  the  results  that 
have  thus  been  obtained,  with  special 
reference  to  their  application  to  class- 
room problems. 

The  text  is  an  amplification  of  a  lecture 
delivered  before  the  Munich  association 
of  gymnasial  teachers,  and  its  primary 
purpose  is  not  to  contribute  to  the  ex- 
perimental investigation  of  fatigue,  but 
to  inform  and  to  interest  teachers. 

The  following  are  among  the  topics  dis- 
cussed :  The  nature  and  forms  of  fatigue, 
the  symptoms  of  fatigue,  the  measure- 
ment of  fatigue  by  physiological  and  by 
psychological  methods,  the  factors  other 
than  fatigue  that  affect  efficiency  of  men- 
tal work— practice,  adaptation,  warming- 
up,  spurts,  enthusiasm,  etc.  —  and  the 
laws  of  fatigue. 

In  considering  the  application  of  these 
laws  to  school-room  problems,  attention 
is  given  to  the  dependence  of  fatigue 
upon  individual  differences,  upon  age, 
puberty,  the  length  of  lesson  periods,  the 
number  of  lessons  per  day,  the  day  of 
the  week,  the  introduction  of  various 
rest  pauses  (recesses,  holidays,  vacations, 
etc.),  change  of  occupation,  the  fatigue 
coefficient  of  the  different  studies,  also  to 
hygienic  arrangement  of  the  school  pro- 
gram and  other  practical  problems.  A 
selected  bibliography  closes  the  mono- 
graph., 

The  translation  is  offered  with  the  con- 
viction that  it  will  meet  a  very  general 
demand  on  the  part  of  the  teacher  of 
educational  psychology  and  of  the  hy- 
giene of  instruction  for  a  clear  and  sys- 
tematic presentation  of  the  problem  of 
mental  fatigue  and  its  relation  to  school 
work. 


WARWICK  &  YOBS,  Inc.,  BALTIMORE,  MD. 


Relative 

Efficiency 

of 

Phonetic 

Alpha- 

bets 


An   Experi- 
mental Inves- 
tigation of 
the   Compara- 
tive Merits  of 
the  Webster 
Key  Alphabet 
and  the 
Proposed  Key 
Alphabet 
Submitted  to 
the   National 
Education 
Association. 


By 
GUY 

MONTROSE 
WHIFFLE, 


Price: 

o,  60  pages. 
35c.    paper 
binding. 

WARWICK 


This  monograph  will  exert  a  two-fold 
appeal  to  those  who  aim  to  keep  abreast 
of  present-day  movements  in  education. 
First,  in  that  it  offers  an  excellent  ex- 
ample of  the  application  of  the  experi- 
mental method  to  a  pedagogical  problem, 
and  in  this  respect  will  take  its  place  as 
a  contribution  to  experimental  pedagogy ; 
secondly,  in  that  it  deals  with  an  im- 
portant topic  just  now  a  matter  of  gen- 
eral discussion  in  educational  circles. 

The  National  Education  Association 
has  under  consideration  the  adoption  of 
a  new  key-alphabet  for  phonetic  nota- 
tion. The  merits  of  the  proposed  alpha- 
bet have  been  the  subject  of  extensive 
and  lively  debate,  but  no  one  has  hither- 
to  done  the  obvious  thing  and  tried  out 
the  new  alphabet  under  experimental 
conditions.  This  Dr.  Whipple  has  ac- 
complished, and  the  results  will  interest 
every  teacher  who  uses  a  phonetic  alpha- 
bet in  his  class  work  as  well  as  every 
educator  who  believes  with  the  author 
that,  in  the  school  as  well  as  in  other 
realms  of  life,  "you  can  tell  by  trying." 

In  view  of  the  fact  that  the  subject  of 
phonetic  alphabets  will  be  given  much 
attention  by  educators  during  the  next 
year,  this  work  is  offered  at  a  price 
which  will  place  it  easily  in  reach  of 
teachers  in  city  and  rural  schools,  and 
also  the  members  of  clubs  and  reading 
circles. 

&  YORK,  Inc.,  BALTIMORE,  MD. 


Back- 
ward and 
Feeble- 
Minded 
Children 

A  Series  of 
Studies  ia 
Clinical 
Psychology. 


By 
EDMUND 


Price: 

12mof 

200  payct, 
illus. 
$1.40. 


Bach  of  the  more  populous  States  has 
several  thousand  mental  defectives,  large 
numbers  of  whom  are  attending  the  pub- 
lic schools.  They  usually  make  little 
progress  and  are  distressingly  disturbing 
factors  in  the  regular  classes.  In  Ger- 
many, and  recently  in  France,  and  in 
some  of  our  own  cities,  these  children 
are  being  placed  in  special  classes  or  in 
special  schools,  according  to  the  degree 
of  defect.  Teachers  and  school  experi- 
ence immediate  relief,  and  the  children 
themselves  are  the  greatest  beneficiaries. 
All  the  schools  have  these  defectives,  and 
the  problem  of  recognizing  and  caring 
for  them  is  an  immediately  pressing  one 
in  all  our  cities,  towns  and  rural  dis- 
tricts. 

Following  a  yew  in  the  clinics  of  Paris, 
Dr.  Huey's  posit  on  at  Lincoln  for  nearly 
a  year  and  a  half  involved  making  a 
mental  examination  of  each  new  ad- 
mission to  this,  one  of  the  largest  state 
institutions  for  the  feeble-minded. 

As  research  psychologist  to  the  Insti- 
tution Dr.  Huey  made  careful  psychologi- 
cal study  of  35  selected  cases  which  rep- 
resent the  transition  zone  between  feeble- 
mindedness and  non-feeble-mindedness. 
These  are  just  the  border  cases  that  puz- 
zle the  school  principal  or  the  clinician. 
In  this  volume  he  presents  case  after 
case  representing  various  types  and 
groups  of  backward  and  feeble-minded 
children.  The  mental  and  physical  char- 
acteristics of  each  child  and  the  salient 
features  of  different  groups  are  clearly 
stated,  with  charts  which  graphically 
present  the  results  of  various  measure- 
ments and  tests. 

The  methods  of  making  examinations 
and  tests  and  of  making  observations  and 
gathering  data  needed  for  the  interpre- 
tation of  any  given  case  are  illustrated 
in  detail.  The  concreteness  of  the  ma- 
terial and  the  abundance  of  illustrative 
examples  will  be  appreciated  by  all,  and 
make  the  studies  intelligible  even  to 
those  unfamiliar  with  psychological 
technique. 


WARWICK  &  YORK,  Inc.,  BALTIMORE,  MD. 


Experi- 
mental 
Studies 
of  Mental 
Defectives 

A  Critique  of 
the   Binet- 
Simon    Tests 
and  a  Contri- 
bution to  the 
Psychology 
of   Epilepsy. 


By 

J.    E. 

WALLACE 
WALLIN, 
Ph.D. 


Alout 

150  pages. 

$1.25. 


The  Binet-Simon  tests  have  been  hailed 
by  popular  writers  and  even  by  some 
scientific  workers  as  a  wonderful  mental 
X-ray  machine,  which  will  enable  us  to 
dissect  the  mental  and  moral  mechan- 
isms of  any  normal  or  abnormal  indi- 
vidual. But  those  who  have  had  ex- 
tensive experience  with  these  tests  know 
that,  despite  their  very  great  practical 
value,  they  have  numerous  imperfections 
and  definite  limitations.  These  imperfec- 
tions and  limitations  can  be  made  known 
only  by  thoroughgoing  trial  on  large 
groups  of  individuals  by  expert  investi- 
gators. Dr.  Wallin  is  well  qualified  by 
training  and  experience  to  undertake  this 
work,  and  he  has  presented  in  this,  the 
seventh  of  the  series  of  Educational 
Psychology  Monographs,  a  systematic 
critical  study  of  the  results  of  the  Binet 
Scale  when  applied  to  a  colony  of  epi- 
leptic children,  and  has  included  a  guide 
for  the  conduct  of  the  tests. 

In  the  course  of  his  study  certain  facts 
have  been  revealed  concerning  the  men- 
tal status  of  the  epileptic  which  should 
interest  the  schoolman  as  well  as  the 
alienist  and  the  physician,  for  epileptic 
children  constitute  a  numerous  class 
which  grades  nearer  the  public  school 
laggard  than  do  feeble-minded  children, 
and  which  cannot  be  reached  by  the  cut- 
and-dried  methods  of  the  schools,  but  re- 
quires a  special  educational  regime. 
Moreover,  epilepsy,  despite  the  investiga- 
tions of  many  alienists,  still  remains  a 
little  understood  pathological  condition 
with  marked  disturbance  of  mentality. 

We  commend  this  contribution  to  the 
attention  of  physicians,  alienists  and  all 
schoolmen  who  are  interested  In  the 
scientific  examination  of  mental  de- 
ficiency. 


WARWICK  &  YORK,  Inc.,  BALTIMORE,  MD. 


Varia- 
tions in 
the 

Grades  of 
High- 
School 
Pupils 


By 

CLARENCE 

TRUMAN 

GRAY. 


Cloth  cat 

120  pages. 

$1.25. 


Ten  years  ago  no  serious  attempt  had 
been  made  to  study  scientifically  the 
relative  merits  of  various  systems  of 
grading  students,  despite  the  fact  that 
statistical  methods  for  undertaking  such 
studies  were  fully  available  and  that 
grading  plays  so  large  a  r61e  in  the 
school  career  of  hundreds  of  thousands 
of  school  children.  In  the  last  five 
years,  however,  this  inviting  field  has 
been  the  scene  of  numerous  important 
investigations,  so  that  we  have  at  least 
arrived  at  a  better  understanding  of  the 
nature  of  the  problem  and  of  the  general 
line  along  which  progress  must  be  made. 

In  the  present  monograph  Mr.  Gray 
reports  the  methods  and  results  of  his 
investigation  of  one  phase  of  the  general 
problem,  viz.,  the  nature,  degree  and 
causes  of  the  variations  occurring  in  the 
grades  of  high-school  pupils.  The  gen- 
eral aim  of  his  study  is  to  base  an  edu- 
cational investigation  upon  school  grades. 
It  is  usually  argued  that  such  marks 
are  inaccurate,  that  they  are  complex, 
that  they  are  not  scientific,  and,  above 
all,  that  it  is  impossible  to  measure 
mental  traits  by  such  cold  statistics  as 
grades  afford.  In  direct  contrast  to 
these  arguments  stands  the  fact  that 
all  promotions  from  the  kindergarten 
through  the  university  are  based  upon 
this  so-called  inaccurate,  complex,  unsci- 
entific and  cold  estimates  of  progress 
and  achievement.  One  of  the  most  vital 
and  fundamental  principles  of  any  school 
system  is  its  plan  of  promotions,  and 
because  of  the  close  relation  between 
promotions  and  grades  there  is  the  most 
urgent  need  that  schoolmen  become  in- 
terested in  the  problems  of  grading. 
Variations  in  the  Grades  of  High-School 
Pupils  should  interest  all  teachers,  and 
more  particularly  all  school  administra- 
tors, because  the  author  not  only  shows 
clearly  how  unreliable  are  the  grades 
commonly  given  by  teachers,  and  makes 
evident  the  need  of  instruction  and  train- 
ing in  grading,  but  also  presents  a  rela- 
tively simple  method  by  means  of  which 
any  high-school  principal  can  study  the 
condition  of  the  grading  in  his  own 
school  and  take  due  steps  to  remedy  the 
faults  that  he  may  find. 


WARWICK  &  YORK,  Inc.,  BALTIMORE,  MD. 


How  I 
Kept  My 
Baby 
Well 


By 

ANNA  Ot. 
NOYES. 


Cloth, 

Illustrated, 

ca,  180  pages. 

$1.25. 
WARWICK 


The  fact  that  the  Journal  of  Educa- 
tional Psychology  has  defined  its  scope 
to  include  the  consideration  of  child  psy- 
chology and  hygiene  justifies  the  inclu- 
sion in  the  allied  series  of  Educational 
Psychology  Monographs  of  the  material 
set  forth  in  the  present  volume. 

Mrs.  Noyes  has  made  a  contribution  of 
real  interest  to  physicians  and  nurses,  to 
mothers  and  fathers,  and  to  students  of 
childhood  generally.  The  value  of  her 
work  is  twofold.  On  the  one  hand,  it 
points  the  way  to  a  method  and  type  of 
observation  that  any  intelligent  mother 
can  undertake  with  profit  to  herself  and 
to  others,  and  in  so  far  disproves  the 
contention  of  some  critics  of  the  child- 
study  movement  that  observations  of 
young  children  by  their  own  mothers  can 
never  yield  data  of  real  value  ;  on  the 
other  hand,  it  furnishes  generalizations 
in  the  shape  of  principles  or  rules  gov- 
erning feeding,  clothing  and  the  general 
control  of  infant  development  that  will 
be  of  direct  utility  to  those  who,  like 
the  author,  face  that  vital  problem- 
how  to  keep  the  baby  well.  Mrs.  Noyes 
has  displayed  commendable  caution  in 
drawing  these  generalizations.  It  is  not 
asserted  that  what  applied  to  her  own 
baby  will  apply  invariably  to  any  other 
baby,  but  only  that  it  undoubtedly  will 
apply  to  many  babies,  and  that  her 
method  of  attacking  the  problem  is,  at 
any  rate,  a  method  that  other  mothers 
may  follow  to  advantage  when  confront- 
ed with  the  same  situation. 

The  conservation  of  human  life  by  the 
reduction  of  infant  mortality  is  a  noble 
undertaking,  and  it  is  hoped  that  this 
little  contribution  may  in  some  measure 
further  that  undertaking. 

The  volume  is  profusely  illustrated. 
The  author  and  Mr.  Noyes  followed  the 
life  of  the  child  through  his  first  two 
years  with  a  camera  just  as  faithfully 
as  the  mother  followed  him  with  her 
charts  and  memorandum  pad.  As  a  con- 
sequence there  appear  as  illustrations 
more  than  sixty  pictures  of  the  baby, 
most  of  them  full-page  cuts.  The  book 
also  contains  some  forty  or  fifty  full- 
page  charts.  Both  photographs  and 
charts  greatly  enhance  the  value  of  the 
book. 

ft  YORK,  Inc.,  BALTIMORE,  MD. 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 
LOAN  DEPT. 

This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 
Renewed  books  are  subject  to  immediate  recall. 


23Jun'59lP 

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